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CO GO a: K 64083 A HISTORY OF MATHEMATICAL NOTATIONS VOLUME I NOTATIONS IN ELEMENTARY MATHEMATICS A HISTORY OF ATHEMATICAL NOTATIONS BY FLORIAN CAJORJ^H.D. Professor of the History of Mathematics University of California VOLUME 1 NOTATIONS IN ELEMENTARY MATHEMATICS THE OPEN COURT COMPANY. PUBLISHERS, 86, STRAND, LONDON, W.C.2. COPYRIGHT 1928 BY THE OPEN COURT PUBLISHING COMPANY Published September 1928 Composed and Printed By The University of Chicago Preu Chicago. Illinois. U.S.A. PREFACE The study of the history of mathematical notations was sug- gested to me by Professor E. H. Moore, of the University of Chicago. To him and to Professor M. W. Haskell, of the University of California, I am indebted for encouragement in the pursuit of this research. As completed in August, 1925, the present history was intended to be brought out in one volume. To Professor H. E. Slaught, of the Uni- versity of Chicago, I owe the suggestion that the work be divided into two volumes, of which the first should limit itself to the history of symbols in elementary mathematics, since such a volume would ap- peal to a wider constituency of readers than would be the case with the part on symbols in higher mathematics. To Professor Slaught I also owe generous and vital assistance in many other ways. He exam- ined the entire manuscript of this work in detail, and brought it to the sympathetic attention of the Open Court Publishing Company. I desire to record my gratitude to Mrs. Mary Hegeler Carus, president of the Open Court Publishing Company, for undertaking this expen- sive publication from which no financial profits can be expected to accrue. I gratefully acknowledge the assistance in the reading of the proofs of part of this history rendered by Professor Haskell, of the Uni- versity of California; Professor R. C. Archibald, of Brown University; and Professor L. C. Karpinski, of the University of Michigan. FLORIAN CAJORI UNIVERSITY OF CALIFORNIA . TABLE OF CONTENTS I. INTRODUCTION PARAGRAPHS II. NUMERAL SYMBOLS AND COMBINATIONS OF SYMBOLS . . . 1-99 Babylonians 1-15 Egyptians 16-26 Phoenicians and Syrians 27-28 Hebrews 29-31 Greeks 32-44 Early Arabs 45 Romans 46-61 Peruvian and North American Knot Records .... 62-65 Aztecs 66-67 Maya 68 Chinese and Japanese 69-73 Hindu- Arabic Numerals 74-99 Introduction 74-77 Principle of Local Value 78-80 Forms of Numerals 81-88 Freak Forms 89 Negative Numerals 90 Grouping of Digits in Numeration 91 The Spanish Calderon 92-93 The Portuguese Cifrao 94 Relative Size of Numerals in Tables 95 Fanciful Hypotheses on the Origin of Numeral Forms . 96 A Sporadic Artificial System 97 General Remarks 98 Opinion of Laplace 99 III. SYMBOLS IN ARITHMETIC AND ALGEBRA (ELEMENTARY PART) 100 A. Groups of Symbols Used by Individual Writers ... 101 Greeks Diophantus, Third Century A.D 101-5 Hindu Brahmagupta, Seventh Century .... 106-8 Hindu The Bakhshal! Manuscript 109 Hindu Bhaskara, Twelfth Century 110-14 Arabic al-Khow&rizmi, Ninth Century .... 115 Arabic al-Karkhf, Eleventh Century 116 Byzantine Michael Psellus, Eleventh Century . . 117 Arabic Ibn Albanna, Thirteenth Century ... 118 Chinese Chu Shih-Chieh, Fourteenth Century . .119, 120 vii viii TABLE OF CONTENTS PARAGRAPHS Byzantine Maximus Planudes, Fourteenth Century 121 Italian Leonardo of Pisa, Thirteenth Century . . 122 French Nicole Oresme, Fourteenth Century . . . 123 Arabic al-Qalasadi, Fifteenth Century .... 124 German Regiomontanus, Fifteenth Century . . . 125-27 ItalianEarliest Printed Arithmetic, 1478 . . . . 128 French Nicolas Chuquet, 1484 129-31 French Estienne de la Roche, 1520 132 Italian Pietro Borgi, 1484, 1488 133 Italian Luca Pacioli, 1494, 1523 134-38 Italian F. Ghaligai, 1521, 1548, 1552 139 Italian H. Cardan, 1532, 1545, 1570 140, 141 Italian Nicolo Tartaglia, 1506-60 142, 143 Italian Rafaele Bombelli, 1572 144, 145 German Johann Widman, 1489, 1526 146 Austrian Grarnrnateus, 1518, 1535 147 German Christoff Rudolff, 1525 148, 149 Dutch Gielis van der Hoecke, 1537 150 German Michael Stifel, 1544, 1545, 1553 .... 151-56 German Nicolaus Copernicus, 1566 157 German Johann Scheubel, 1545, 1551 .... 158, 159 Maltese Wil. Klebitius, 1565 160 German Christophorus Clavius, 1608 161 Belgium Simon Stevin, 1585 162, 163 Lorraine Albert Girard, 1629 164 German-SpanishMarco Aurel, 1552 165 Portuguese-Spanish Pedro Nunez, 1567 .... 166 English Robert Recorde, 1543(?), 1557 .... 167-68 English John Dee, 1570 169 English Leonard and Thomas Digges, 1579 . . . 170 English Thomas Mastcrson, 1592 171 French Jacques Peletier, 1554 172 French Jean Buteon, 1559 173 French Guillaume Gosselin, 1577 174 French Francis Vieta, 1591 176-78 Italian Bonaventura Cavalieri, 1647 179 English William Oughtred, 1631, 1632, 1657 . . . 180-87 English Thomas Harriot, 1631 188 French Pierre HSrigone, 1634, 1644 189 Scot-FrenchJames Hume, 1635, 1636 .... 190 French Rene* Descartes 191 English Isaac Barrow 192 English Richard Rawlinson, 1655-68 193 Swiss Johann Heinrich Rahn 194 TABLE OF CONTENTS ix PARAGRAPHS English John Wallis, 1655, 1657, 1685 .... 195, 196 Extract from Ada eruditorum, Leipzig, 1708 . . . 197 Extract from Miscellanea Berolinensia, 1710 (Duo to G. W. Leibniz) 198 Conclusions 199 B. Topical Survey of the Use of Notations 200-356 Signs of Addition and Subtraction 200-216 Early Symbols 200 Origin and Meaning of the Signs 201-3 Spread of the + and Symbols 204 Shapes of the + Sign 205-7 Varieties of - Signs 208, 209 Symbols for " Plus or Minus" 210,211 Certain Other Specialized Uses of + and . . 212-14 Four Unusual Signs 215 Composition of Ratios 216 Signs of Multiplication 217-34 Early Symbols 217 Early Uses of the St. Andrew's Cross, but Not as the Symbol of Multiplication of Two Numbers . . 218-30 The Process of Two False Positions .... 219 Compound Proportions with Integers .... 220 Proportions Involving Fractions 221 Addition and Subtraction of Fractions . . . 222 Division of Fractions 223 Casting Out the 9's, 7's, or ll's 225 Multiplication of Integers 226 Reducing Radicals to Radicals of the Same Order 227 Marking the Place for " Thousands" .... 228 Place of Multiplication Table above 5X5 . . 229 The St. Andrew's Cross Used as a Symbol of Multi- plication 231 Unsuccessful Symbols for Multiplication . . . 232 The Dot for Multiplication 233 The St. Andrew's Cross in Notation for Transfinite Ordinal Numbers 234 Signs of Division and Ratio 235-47 Early Symbols 235,236 Rahn's Notation 237 Leibniz's Notations 238 Relative Position of Divisor and Dividend ... 241 Order of Operations in Terms Containing Both -f- and X 242 A Critical Estimate of : and - as Symbols . . 243 TABLE OF CONTENTS PABAQRAPH8 Notations for Geometric Ratio 244 Division in the Algebra of Complex Numbers . . 247 Signs of Proportion 248-50 Arithmetical and Geometrical Progression . . . 248 Arithmetical Proportion 249 Geometrical Proportion 250 OughtrecTs Notation 251 Struggle in England between Oughtred's and Wing's Notations before 1700 252 Struggle in England between Oughtred's and Wing's Notations during 1700-1750 253 Sporadic Notations 254 Oughtred's Notation on the European Continent . 255 Slight Modifications of Oughtred's Notation . . 257 The Notation : : : : in Europe and America . . 258 The Notation of Leibniz 259 Signs of Equality 260-70 Early Symbols 260 Recorde's Sign of Equality 261 Different Meanings of = 262 Competing Symbols 263 Descartes' Sign of Equality 264 Variations in the Form of Descartes' Symbol . . 265 Struggle for Supremacy 266 Variation in the Form of Recorde's Symbol . . . 268 Variation in the Manner of Using It 269 Nearly Equal 270 Signs of Common Fractions 271-75 Early Forms 271 The Fractional Line 272 Special Symbols for Simple Fractions 274 TheSolidus 275 Signs of Decimal Fractions 276-89 Stevin's Notation 276 Other Notations Used before 1617 278 Did Pitiscus Use the Decimal Point? .... 279 Decimal Comma and Point of Napier .... 282 Seventeenth-Century Notations Used after 1617 . 283 Eighteenth-Century Discard of Clumsy Notations . 285 Nineteenth Century : Different Positions for Point and for Comma 286 Signs for Repeating Decimals 289 Signs of Powers 290-315 General Remarks 290 TABLE OP CONTENTS PARAGRAPHS Double Significance of R and I 291 Facsimiles of Symbols in Manuscripts .... 293 Two General Plans for Marking Powers .... 294 Early Symbolisms: Abbreviative Plan, Index Plan 295 Notations Applied Only to an Unknown Quantity, the Base Being Omitted 296 Notations Applied to Any Quantity, the Base Being Designated 297 Descartes' Notation of 1637 298 Did Stampioen Arrive at Descartes' Notation Inde- pendently? 299 Notations Used by Descartes before 1637 . . . 300 Use of H6rigone's Notation after 1637 .... 301 Later Use of Hume's Notation of 1636 .... 302 Other Exponential Notations Suggested after 1637 . 303 Spread of Descartes' Notation 307 Negative, Fractional, and Literal Exponents . . 308 Imaginary Exponents 309 Notation for Principal Values 312 Complicated Exponents 313 D. F. Gregory's (+) r 314 Conclusions , 315 Signs for Roots 316-38 Early Forms, General Statement 316, 317 The Sign $, First Appearance 318 Sixteenth-Century Use of /J 319 Seventeenth-Century Use of # 321 The Sign I 322 Napier's Line Symbolism 323 The Sign V 324-38 Origin of V 324 Spread of the V 327 Rudolff's Signs outside of Germany .... 328 Stevin's Numeral Root-Indices ...... 329 Rudolff and Stifel's Aggregation Signs . . . 332 Descartes' Union of Radical Sign and Vinculum . 333 Other Signs of Aggregation of Terms . . .. . 334 Redundancy in the Use of Aggregation Signs . 335 Peculiar Dutch Symbolism 336 Principal Root- Values 337 Recommendation of the U.S. National Committee 338 Signs for Unknown Numbers 339-41 Early Forms 339 xii TABLE OF CONTENTS PARAGRAPHS Crossed Numerals Representing Powers of Un- knowns . 340 Descartes' 2, y, x 340 Spread of Descartes' Signs 341 Signs of Aggregation 342-56 Introduction 342 Aggregation Expressed by Letters 343 Aggregation Expressed by Horizontal Bars or Vincu- lums 344 Aggregation Expressed by Dots 348 Aggregation Expressed by Commas 349 Aggregation Expressed by Parentheses .... 350 Early Occurrence of Parentheses 351 Terms in an Aggregate Placed in a Verbal Column 353 Marking Binomial Coefficients 354 Special Uses of Parentheses 355 A Star to Mark the Absence of Terms .... 356 IV. SYMBOLS IN GEOMETRY (ELEMENTARY PART) 357-85 A, Ordinary Elementary Geometry 357 Early Use of Pictographs 357 Signs for Angles 360 Signs f or " Perpendicular" 364 Signs for Triangle, Square, Rectangle, Paiiillclogram . 365 The Square as an Operator 366 Sign for Circle 367 Signs for Parallel Lines 368 Signs for Equal and Parallel 369 Signs for Arcs of Circles 370 Other Pictographs 371 Signs for Similarity and Congruence 372 The Sign O for Equivalence 375 Lettering of Geometric Figures 376 Sign for Spherical Excess 380 Symbols in the Statement of Theorems 381 Signs for Incommensurables 382 Unusual Ideographs in Elementary Geometry . . . 383 Algebraic Symbols in Elementary Geometry . . . 384 B. Past Struggles between Symbolists and Rhetoricians in Elementary Geometry .385 INDEX ILLUSTRATIONS FIQURB PARAGRAPHS 1. BABYLONIAN TABLETS OF NIPPUR 4 2. PRINCIPLE OF SUBTRACTION IN BABYLONIAN NUMERALS ... 9 3. BABYLONIAN LUNAR TABLES 11 4. MATHEMATICAL CUNEIFORM TABLET CBS 8536 IN THE MUSEUM OF THE UNIVERSITY OF PENNSYLVANIA 11 5. EGYPTIAN NUMERALS 17 6. EGYPTIAN SYMBOLISM FOR SIMPLE FRACTIONS 18 7. ALGEBRAIC EQUATION IN AHMES 23 8. HIEROGLYPHIC, HIERATIC, AND COPTIC NUMERALS 24 9. PALMYRA (SYRIA) NUMERALS 27 10. SYRIAN NUMERALS 28 11. HEBREW NUMERALS 30 12. COMPUTING TABLE OF SALAMIS 36 13. ACCOUNT OF DISBURSEMENTS OF THE ATHENIAN STATE, 418- 415 B.C. 36 14. ARABIC ALPHABETIC NUMERALS 45 15. DEGENERATE FORMS OF ROMAN NUMERALS 56 16. QUIPU FROM ANCIENT CHANCAY IN PERU 65 17. DIAGRAM OF THE Two RIGHT-HAND GROUPS 65 18. AZTEC NUMERALS 66 19. DRESDEN CODEX OF MAYA 67 20. EARLY CHINESE KNOTS IN STRINGS, REPRESENTING NUMERALS . 70 21. CHINESE AND JAPANESE NUMERALS 74 22. HILL'S TABLE OF BOETHIAN APICES 80 23. TABLE OF IMPORTANT NUMERAL FORMS 80 24. OLD ARABIC AND HINDU-ARABIC NUMERALS 83 25. NUMERALS OF THE MONK NEOPHYTOS 88 26. CHR. RUDOLFF'S NUMERALS AND FRACTIONS 89 27. A CONTRACT, MEXICO CITY, 1649 93 xiv ILLUSTRATIONS FIGURE PARAGRAPHS 28. REAL ESTATE SALE, MEXICO CITY, 1718 . 94 29. FANCIFUL HYPOTHESES 96 30. NUMERALS DESCRIBED BY NOVIOMAGUS 98 31. SANSKRIT SYMBOLS FOR THE UNKNOWN 108 32. BAKHSHALI ARITHMETIC 109 33. SRIDHARA'S Trisdtika 112 34. ORESME'S Algorismus Proportionum 123 35. AL-QALASADI'S ALGEBRAIC SYMBOLS 125 36. COMPUTATIONS OF REGIOMONTANUS 127 37. CALENDAR OF REGIOMONTANUS 128 38. FROM EARLIEST PRINTED ARITHMETIC 128 39. MULTIPLICATIONS IN THE" TREVISO" ARITHMETIC 128 40. DE LA ROCHE'S Larismethique, FOLIO 605 132 41. DE LA ROCHE'S Larismethique, FOLIO 66A 132 42. PART OF PAGE IN PACIOLI'S Summa, 1523 138 43. MARGIN OF FOLIO 1235 IN PACIOLI'S Summa 139 44. PART OF FOLIO 72 OF GHALIGAI'S Practica d'arithmetica, 1552 . 139 45. GHALIGAI'S Practica d'arithmetica, FOLIO 198 139 46. CARDAN, Ars magna, ED. 1663, PAGE 255 141 47. CARDAN, Ars magna, ED. 1663, PAGE 297 141 48. FROM TARTAGLIA'S General Trattato, 1560 143 49. FROM TARTAGLIA'S General Trattato, FOLIO 4 144 50. FROM BOMBELLI'S Algebra, 1572 144 51. BOMBELLI'S Algebra (1579 IMPRESSION), PAGE 161 .... 145 52. FROM THE MS OF BOMBELLI'S Algebra IN THE LIBRARY OF BOLOGNA 145 53. FROM PAMPHLET No. 595AT IN THE LIBRARY OF THE UNIVERSITY OF BOLOGNA 146 54. WIDMAN'S Rechnung, 1526 146 55. FROM THE ARITHMETIC OF GRAMMATEUS 146 56. FROM THE ARITHMETIC OF GRAMMATEUS, 1535 147 57. FROM THE ARITHMETIC OF GRAMMATEUS, 1518(?) 147 58. FROM CHR. RUDOLFF'S Coss, 1525 148 ILLUSTRATIONS xv PARAGRAPHS 59. FROM CHR. RUDOLFF'S Coss, Ev 148 ' 60. FROM VAN DER HOECKE' In arithmetica 150 61. PART OF PAGE FROM STIFEL'S Arithmetica intcgra, 1544 . . . 150 62. FROM STIFEL'S Arithmetica Integra, FOLIO 31B 152 63. FROM STIFEL'S EDITION OF RUDOLFF'S Coss, 1553 156 64. SCHEUBEL, INTRODUCTION TO EUCLID, PAGE 28 159 65. W. KLEBITIUS, BOOKLET, 1565 161 66. FROM GLAVIUS' Algebra, 1608 161 67. FROM S. STEVIN'S Le Thiende, 1585 162 68. FROM S. STEVIN'S Arithmetiqve 162 69. FROM S. STEVIN'S Arithmetiqve 164 70. FROM AUREL'S Arithmetica 165 71. R. RECORDS, Whetstone of Witte, 1557 168 72. FRACTIONS IN RECORDE 168 73. RADICALS IN RECORDE 168 74. RADICALS IN DEE'S PREFACE 169 75. PROPORTION IN DEE'S PREFACE 169 76. FROM DIGGES'S Stratioticos 170 77. EQUATIONS IN DIGGES 172 78. EQUALITY IN DIGGES 172 79. FROM THOMAS MASTERSON'S Arithrneticke, 1592 172 80. J. PELETIER'S Algebra, 1554 172 81. ALGEBRAIC OPERATIONS IN PELETIER'S Algebra 172 82. FROM J. BUTEON, Arithmetica, 1559 173 83. GOSSELIN'S De arte magna, 1577 174 84. VIETA, In artem analyticam, 1591 176 85. VIETA, De emendatione aeqvationvm 178 86. B. CAVALIERI, Exercitationes, 1647 179 87. FROM THOMAS HARRIOT, 1631, PAGE 101 189 88. FROM THOMAS HARRIOT, 1631, PAGE 65 189 89. FROM HERIGONE, Cursus mathematicus, 1644 189 90. ROMAN NUMERALS FOR x IN J. HUME, 1635 191 xvi ILLUSTRATIONS FIGURE PARAGRAPHS 91. RADICALS IN J. HUME, 1635 191 92. R. DESCARTES, Gtomttrie 191 93. I. BARROW'S Euclid, LATIN EDITION. NOTES BY ISAAC NEWTON . 193 94. I. BARROW'S Ewlid, ENGLISH EDITION 193 95. RICH. RAWLINSON'S SYMBOLS 194 96. RAHN'S Teutsche Algebra, 1659 195 97. BRANCKER'S TRANSLATION OF RAHN, 1668 195 98. J. WALLIS, 1657 195 99. FROM THE HIEROGLYPHIC TRANSLATION OF THE AHMES PAPYRUS 200 100. MINUS SIGN IN THE GERMAN MS C. 80, DRESDEN LIBRARY . . 201 101. PLUS AND MINUS SIGNS IN THE LATIN MS C. 80, DRESDEN LIBRARY 201 102. WIDMANS' MARGINAL NOTE TO MS C. 80, DRESDEN LIBRARY . 201 103. FROM THE ARITHMETIC OF BOETHIUS, 1488 250 104. SIGNS IN GERMAN MSS AND EARLY GERMAN BOOKS .... 294 105. WRITTEN ALGEBRAIC SYMBOLS FOR POWERS FROM PEREZ DE MOYA'S Arithmetica 294 106. E. WARING'S REPEATED EXPONENTS 313 INTRODUCTION In this history it has been an aim to give not only the first appear- ance of a symbol and its origin (whenever possible), but also to indi- cate the competition encountered and the spread of the symbol among writers in different countries. It is the latter part of our program which has given bulk to this history. The rise of certain symbols, their day of popularity, and their eventual decline constitute in many cases an interesting story. Our endeavor has been to do justice to obsolete and obsolescent notations, as well as to those which have survived and enjoy the favor of mathe- maticians of the present moment. If the object of this history of notations were simply to present an array of facts, more or less interesting to some students of mathe- matics if, in other words, this undertaking had no ulterior motive then indeed the wisdom of preparing and publishing so large a book might be questioned. But the author believes that this history consti- tutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of the notational problems of the present time. | n NUMERAL SYMBOLS AND COMBINATIONS OF SYMBOLS BABYLONIANS 1. In the Babylonian notation of numbers a vertical wedge Y stood for 1, while the characters ^ and Y>- signified 10 and 100, respectively. Grotefend 1 believes the character for 10 originally to have been the picture of two hands, as held in prayer, the palms being pressed together, the fingers close to each other, but the thumbs thrust out. Ordinarily, two principles were employed in the Babylonial no- tation the additive and multiplicative. We shall see that limited use was made of a third principle, that of subtraction. 2. Numbers below 200 were expressed ordinarily by symbols whose respective values were to be added. Thus, Y^XKYYY stands for 123. The principle of multiplication reveals itself in < |>- where the smaller symbol 10, placed before the 100, is to be multiplied by 100, so that this symbolism designates 1,000. 3. These cuneiform symbols were probably invented by the early Sumerians. Their inscriptions disclose the use of a decimal scale of numbers and also of a sexagesimal scale. 2 Early Sumerian clay tablets contain also numerals expressed by circles and curved signs, made with the blunt circular end of a stylus, the ordinary wedge-shaped characters being made with the pointed end. A circle stood for 10, a semicircular or lunar sign stood for 1. Thus, a "round-up" of cattle shows J*DDD> or ^ cows. 3 4. The sexagesimal scale was first discovered on a tablet by E. Hincks 4 in 1854. It records the magnitude of the illuminated portion 1 His first papers appeared in Gottingische Gelehrte Anzeigen (1802), Stuck 149 und 178; ibid. (1803), Stuck 60 und 117. 2 In the division of the year and of the day, the Babylonians used also the duodecimal plan. 8 G. A. Barton, Haverford Library Collection of Tablets, Part I (Philadelphia, 1905), Plate 3, HCL 17, obverse; see also Plates 20, 26, 34, 35. Allotte de la Fuye, "En-e-tar-zi pate*si de Lagas," H. V. Hilprecht Anniversary Volume (Chi- cago, 1909), p. 128, 133. 4 "On the Assyrian Mythology," Transactions of the Royal Irish Academy. "Polite Literature," Vol. XXII, Part 6 (Dublin, 1855), p. 406, 407. 2 OLD NUMERAL SYMBOLS 3 of the moon's disk for every day from new to full moon, the whole disk being assumed to consist of 240 parts. The illuminated parts during the first five days are the series 5, 10, 20, 40, 1.20, which is a geo- metrical progression, on the assumption that the last number is 80. From here on the series becomes arithmetical, 1.20, 1.36, 1.52, 2.8, 2.24, 2.40, 2.56, 3.12, 3.28, 3.44, 4, the common difference being 16. The last number is written in the tablet X^, and, according to Hincks's interpretation, stood for 4 X 60 = 240. Obverse. Reverse. FIG. 1. Babylonian tablets from Nippur, about 2400 B.C. 5, Hincks's explanation was confirmed by the decipherment of tablets found at Senkereh, near Babylon, in 1854, and called the Tab- lets of Senkereh. One tablet was found to contain a table of square numbers, from I 2 to 60 2 , a second one a table of cube numbers from I 3 to 32 3 . The tablets were probably written between 2300 and 1600 B.C. Various scholars contributed toward their interpretation. Among them \vere George Smith (1872), J. Oppert, Sir H. Rawlinson, Fr. Lenormant, and finally R. Lepsius. 1 The numbers 1, 4, 9, 16, 25, 36, George Smith, North British Review (July, 1870), p. 332 n.; J. Oppert, Journal asiatique (August-September, 1872; October-November, 1874); J. Oppert, talon des tnesures assyr. fixe" par les textes cuneiformes (Paris, 1874) ; Sir H. Rawlinson and G. Smith, "The Cuneiform Inscriptions of Western Asia," Vol. IV: A Selection from the Miscellaneous Inscriptions of Assyria (London, 1875), Plate 40; R. Lepsius, "Die Babylonisch-Assyrischen Langenmaasse nach der Tafel von Senkereh," Abhandlungen der Koniglichen Akademie der Wissen- schaften zu Berlin (aus dem Jahre 1877 [Berlin, 1878], Philosophisch-historische Klasse), p. 105-44. 4 A HISTORY OF MATHEMATICAL NOTATIONS and 49 are given as the squares of the first seven integers, respecti We have next 1.4 = 8 2 , 1.21 = 9 2 , 1.40= 10 2 , etc. This clearly indi the use of the sexagesimal scale which makes 1.4 = 60+4, 1.21 = 21. 1.40 = 60+40, etc. This sexagesimal system marks the ea: appearance of the all-important "principle of position" in wr numbers. In its general and systematic application, this principl quires a symbol for zero. But no such symbol has been found on < Babylonian tablets; records of about 200 B.C. give a symbol for as we shall see later, but it was not used in calculation. The ea: thorough and systematic application of a symbol for zero anc principle of position was made by the Maya of Central America, a the beginning of the Christian Era. 6. An extension of our knowledge of Babylonian mathem was made by H. V. Hilprecht who made excavations at Nuffar ancient Nippur). We reproduce one of his tablets 1 in Figure 1. Hilprecht's transliteration, as given on page 28 of his te as follows: Line 1. 125 720 Line 9. 2,000 Line 2. IGI-GAL-BI 103,680 Line 10. IGI-GAL-BI ( Line 3. 250 360 Line 11. 4,000 Line 4. IGI-GAL-BI 51,840 Line 12. IGI-GAL-BI Line 5. 500 180 Line 13. 8,000 Line 6. IGI-GAL-BI 25,920 Line 14. IGI-GAL-BI Line 7. 1,000 90 Line 15. 16,000 Line 8. IGI-GAL-BI 12,960 Line 16. IGI-GAL-BI 7. In further explanation, observe that in Line 1. 125 = 2X60+5, 720 = 12X60+0 Line 2. Its denominator, 103,680 = [28 X60+48(?)]X 6 Line 3. 250 = 4X60+10, 360 = 6X60+0 Line 4. Its denominator, 51,840 = [14 X 60+24] X60+( Line 5. 500 = 8X60+20, 180 = 3X60+0 Line 6. Its denominator, 25,920 = [7 X 60+ 12] X 60+0 Line 7. 1,000=16X60+40, 90=1X60+30 Line 8. Its denominator, 12,960 = [3X60+36]X60+0 1 The Babylonian Expedition of the University of Pennsylvania. Seri "Cuneiform Texts," Vol. XX, Part 1, Mathematical, Metrological and C) logical Tablets from the Temple Library of Nippur (Philadelphia, 1906), Pla No. 25. OLD NUMERAL SYMBOLS 5 Line 9. 2,000 = 33X60+20, 18=10+8 Line 10. Its denominator, 6,480 = [IX 60+48] X 60+0 Line 11. 4,000 = [1X60+6]X60+40, 9 Line 12. Its denominator, 3,240 = 54 X 60+0 Line 13. 8,000 = [2X60+13]X60+20, 18 Line 14. Its denominator, 1,620 = 27X60+0 Line 15. 16,000 = [4X60+ 26] X 60+40, 9 Line 16. Its denominator, 810=13X60+30 IGI-GAL = Denominator, / = Its, i.e., the number 12,960,000 or 60 4 . We quote from Hilprecht (op. cit., pp. 28-30): "We observe (a) that the first numbers of all the odd lines (1, 3, 5, 7, 9, 11, 13, 15) form an increasing, and all the numbers of the even lines (preceded by IGI-GAL-BI = ( its denominator') a descending geometrical progression; (6) that the first number of every odd line can be expressed by a fraction which has 12,960,000 as its numerator and the closing number of the corresponding even line as its denomi- nator, in other words, 10 12,960,000 . 12,960,000 . Knn _ 12,960,000 1 ^ ) "" 103,680 ' 51,840 ' 25,920 ' nn 12,960,000 . 9 mn 12,960,000 . 12, > m = -l2W- ' 2 > 00= 6480 ' 4 ' 000== 3 960,000 6,480 ' ' 3,240 ' 12,960,000 . 12,960,000 8,000= lj62Q , 16,000-^. But the closing numbers of all the odd lines (720, 360, 180, 90, 18, 9, 18, 9) are still obscure to me ..... "The question arises, what is the meaning of all this? What in par- ticular is the meaning of the number 12,960,000 ( = 60 4 or 3,600 2 ) which underlies all the mathematical texts here treated ....?.... This ' geometrical number ' (12,960,000), which he [Plato in his Repub- lic viii. 546#-D] calls 'the lord of better and worse births/ is the arithmetical expression of a great law controlling the Universe. According to Adam this law is 'the Law of Change, that law of in- evitable degeneration to which the Universe and all its parts are sub- ject' an interpretation from which I arn obliged to differ. On the contrary, it is the Law of Uniformity or Harmony, i.e. that funda- mental law which governs the Universe and all its parts, and which cannot be ignored and violated without causing an anomaly, i.e. with- out resulting in a degeneration of the race." The nature of the "Pla- tonic number" is still a debated question. 6 A HISTORY OF MATHEMATICAL NOTATIONS 8. In the reading of numbers expressed in the Babylonian sexa- gesimal system, uncertainty arises from the fact that the early Baby- lonians had no symbol for zero. In the foregoing tablets, how do we know, for example, that the last number in the first line is 720 and not 12? Nothing in the symbolism indicates that the 12 is in the place where the local value is "sixties" and not "units." Only from the study of the entire tablet has it been inferred that the number in- tended is 12X60 rather than 12 itself. Sometimes a horizontal line was drawn following a number, apparently to indicate the absence of units of lower denomination. But this procedure was not regular, nor carried on in a manner that indicates the number of vacant places. 9. To avoid confusion some Babylonian documents even in early times contained symbols for 1, 60, 3,600, 216,000, also for 10, 600, 36,000.* Thus was 10, was 3,600, was 36,000. in view of other variants occurring in fchc mathematical tablets from Nippur, notably the numerous variants of "19," 1 some of which may be merely scribal errors : They evidently all go back to the form <^}~ ^^^f (20 1 = 19). FIG. 2. Showing application of the principle of subtraction 10. Besides the principles of addition and multiplication, Baby- lonian tablets reveal also the use of the principle of subtraction, which is familiar to us in the Roman notation XIX (201) for the number 19. Hilprecht has collected ideograms from the Babylonian tablets which he has studied, which represent the number 19. We reproduce his symbols in Figure 2. In each of these twelve ideograms (Fig. 2), the' two symbols to the left signify together 20. Of the symbols im- mediately to the right of the 20, one vertical wedge stands for "one" and the remaining symbols, for instance Y^, for LAL or "minus"; the entire ideogram represents in each of the twelve cases the number 20- lor 19. One finds the principle of subtraction used also with curved signs; 2 D Y*~~D meant 60+20 1, or 79. 1 See Frangois Thureau-Dangin, Recherches sur Vorigine de Vecriture cuntiforme (Paris, 1898), Nos. 485-91, 509-13. See also G. A. Barton, Haverford College Library Collection of Cuneiform Tablets, Part I (Philadelphia, 1905), where the forms are somewhat different; also the Hilprecht Anniversary Volume (Chicago, 1909), p. 128 ff. 2 G. A. Barton, op. cit. t Plate 3, obverse. OLD NUMERAL SYMBOLS 11. The symbol used about the second century B.C. to designate 5 absence of a number, or a blank space, is shown in Figure 3, con- ning numerical data relating to the moon. 1 As previously stated, s symbol, ^ , was not used in computation and therefore performed FIG. 3. Babylonian lunar tables, reverse; full moon for one year, about the 1 of the second century B.C. ly a small part of the functions of our modern zero. The symbol is jn in the tablet in row 10, column 12; also in row 8, column 13. igler's translation of the tablet, given in his book, page 42, is shown low. Of the last column only an indistinct fragment is preserved; 3 rest is broken off. REVERSE Niaannu 2856'30" 1916' " Librae 3 Z 645' 4 i 74 ii 10 ur sik Airu 28 38 30 175430 Scorpii 321 28 620 30 sik Simannu 28 20 30 16 15 Arcitenentia 3 31 39 345 30 sik Dti,zu 28 18 30 14 33 30 Capri 33441 1 10 30 sik Abu 28 36 30 13 9 Aquarii 32756 1 24 30 bar Ululu 29 54 30 13 3 30 Piscium 3 1534 1 59 30 num TiSrltu 29 12 30 11 16 Arietis 258 3 4 34 30 num Araty-s. 29 30 30 10 46 30 Tauri 24054 6 10 num Kishmu 29 48 30 10 35 Geminorurn 2 29 29 3 25 10 num Tebitu 29 57 30 10 32 30 Cancri 22430 067 10 num Sabatu 29 39 30 10 12 Leonis 2 30 53 1 44 50 bar Addru I 29 21 30 9 33 30 Virginis 2 42 56 2 19 50 sik Ad6.ru II 29 330 8 36 Librae 3 021 464 50 sik Nisannu 28 45 30 7 21 30 Scorpii 3 1736 539 50 sik 1 Franz Xaver Kugler, S. J., Die babylonische Mondrechnung (Freiburg im Breis- j, 1900), Plate IV, No. 99 (81-7-6), lower part. A HISTORY OF MATHEMATICAL NOTATIONS OBVERSE W^r- 4 ^r EMte- _^MN w^ W*- s^- 31 ^^- (THff^- El^Si^ ^^ r-w-'n..*' ^teh'^tfilii FIG. 4. Mathematical cuneiform tablet, CBS 8536, in the Museum of the University of Pennsylvania. OLD NUMERAL SYMBOLS 9 12. J. Oppert pointed out the Babylonian use of a designation the sixths, viz., |, , -, f, $. These are unit fractions or fractions ose numerators are one less than the denominators. 1 He also ad- iced evidence pointing to the Babylonian use of sexagesimal frac- is and the use of the sexagesimal system in weights and measures. B occurrence of sexagesimal fractions is shown in tablets recently mined. We reproduce in Figure 4 two out of twelve columns found a tablet described by H. F. Lutz. 2 According to Lutz, the tablet innot be placed later than the Cassite period, but it seems more prob- e that it goes back even to the First Dynasty period, ca. 2000 B.C." 13, To mathematicians the tablet is of interest because it reveals orations with sexagesimal fractions resembling modern operations h decimal fractions. For example, 60 is divided by 81 and the >tient expressed sexagesimally. Again, a sexagesimal number with > fractional places, 44 (26) (40), is multiplied by itself, yielding a .duct in four fractional places, namely, [32]55(18)(31)(6)(40). In 3 notation the [32] stands for 32X60 units, and to the (18), (31), , (40) must be assigned, respectively, the denominators 60, 60 2 , , 60 4 . The tablet contains twelve columns of figures. The first column g. 4) gives the results of dividing 60 in succession by twenty-nine 'erent divisors from 2 to 81. The eleven other columns contain les of multiplication; each of the numbers 50, 48, 45, 44 (26) (40), 36, 30, 25, 24, 22(30), 20 is multiplied by integers up to 20, then by numbers 30, 40, 50, and finally by itself. Using our modern nu- rals, we interpret on page 10 the first and the fifth columns. They dbit a larger number of fractions than do the other columns. e Babylonians had no mark separating the fractional from the in- ral parts of a number. Hence a number like 44 (26) (40) might be 3rpreted in different ways; among the possible meanings are 44 X +26X60+40, 44X60+26+40X60- 1 , and 44+26X60~ x +40X -2. Which interpretation is the correct one can be judged only by context, if at all. The exact meaning of the first two lines in the first column is un- fcain. In this column 60 is divided by each of the integers written the left. The respective quotients are placed on the right. 1 Symbols for such fractions are reproduced also by Thureau-Dangin, op. cit., \. 481-84, 492-508, and by G. A. Barton, Haverford College Library Collection Cuneiform Tablets, Part I (Philadelphia, 1905). 2 "A Mathematical Cuneiform Tablet/' American Journal of Semitic Lan- ges and Literatures, Vol. XXXVI (1920), p. 24^-57. 10 A HISTORY OF MATHEMATICAL NOTATIONS In the fifth column the multiplicand is 44 (26) (40) or 44 jj. The last two lines seem to mean "60 2 -r-44(26)(40) = 81, 60 2 -r81 = 44(26)(40)." First Column .... gal (?) -bi 40 -&m Su a- na gal-bi 30 -am igi 2 30 igi 3 20 igi 4 15 igi 5 12 igi 6 10 igi 8 7(30) igi 9 6(40) igi 10 6 igi 12 5 igi 15 4 igi 16 3(45) igi 18 3(20) igi 20 3 igi 24 2(30) igi 25 2(24) igi 28* 2(13) (20) igi 30 2 igi 35* 1(52) (30) igi 36 1(40) igi 40 1(30) igi 45 1(20) igi 48 1(15) igi 50 1(12) igi 54 1(6) (40) igi 60 1 igi 64 (56) (15) igi 72 (50) igi 80 (45) igi 81 (44) (26) (40) 1 2 3 4 5 6 7 9 10 11 12 13 14 15 16 17 18 19 20 30 40 50 Fifth Column 44(26) (40) 44(26) (40) [1]28(53)(20) [2]13(20) [2]48(56)(40)* [3]42(13)(20) [4]26(40) [6]40 [7]24(26)(40) [8]8(53)(20) [8]53(20) [9]27(46)(40)* [10]22(13)(20) [H]6(40) [12]35(33)(20) [13J20 [14]4(26)(40) [14]48(53)(20) [22]13(20) [29]37(46)(40) [38]2(13)(20)* 44(26) (40)a-na 44(26) (40) [32]55(18)(31)(6)(40) 44 (26) (40) square igi 44(26)(40) 81 igiSl 44 (26) (40) Numbers that are incorrect are marked by an asterisk (*). 14. The Babylonian use of sexagesimal fractions is shown also in a clay tablet described by A. Ungnad. 1 In it the diagonal of a rec- tangle whose sides are 40 and 10 is computed by the approximation 1 Orientalische Literaturzeitung (ed. Peise, 1916), Vol. XIX, p. 363-68. See also Bruno Meissner, Babylonien und Assyrien (Heidelberg, 1925), Vol. II, p. 393. OLD NUMERAL SYMBOLS 11 40+2X40Xl0 2 -h60 2 , yielding 42(13)(20), and also by the approxi- mation 40+10 2 4- 12X401, yielding 41(15). Translated into the deci- mal scale, the first answer is 42.22+, the second is 41.25, the true value being 41.23+. These computations are difficult to explain, except on the assumption that they involve sexagesimal fractions. 15. From what has been said it appears that the Babylonians had ideograms which, transliterated, are Igi-Gal for "denominator" or "division," and Lai for "minus." They had also ideograms which, transliterated, are Igi-Dua for "division," and A-Du and Ara for "times," as in Ara- 1 18, for "1X18 = 18," Ara- 2 36 for "2 X 18 = 36" ; the Ara was used also in "squaring," as in 3 Ara 3 9 for "3X3 = 9." They had the ideogram Ba-Di-E for "cubing," as in 21 -E 3 Ba-Di-E for "3 3 = 27"; also Ib-Di for "square," as in 9-# 3 Ib-Di for "3 2 = 9." The sign A -An rendered numbers "distribu- tive." 1 EGYPTIANS 16. The Egyptian number system is based on the scale of 10, al- though traces of other systems, based on the scales of 5, 12, 20, and 60, are believed to have been discovered. 2 There are three forms of Egyptian numerals: the hieroglyphic, hieratic, and demotic. Of these the hieroglyphic has been traced back to about 3300 B.C. ; 3 it is found mainly on monuments of stone, wood, or metal. Out of the hiero- glyphic sprang a more cursive writing known to us as hieratic. In the beginning the hieratic was simply the hieroglyphic in the rounded forms resulting from the rapid manipulation of a reed-pen as con- trasted with the angular and precise shapes arising from the use of the chisel. About the eighth century B.C. the demotic evolved as a more abbreviated form of cursive writing. It was used since that time down to the beginning of the Christian Era. The important mathematical documents of ancient Egypt were written on papyrus and made use of the hieratic numerals. 4 1 Hilprecht. op. til., p. 23; Arno Poebel, Grundzuge der sumerischen Grammatik (Rostock, 1923), p. 115; B. Meissner, op. cit., p. 387-89. 2 Kurt Sethe, Von ZdhLen und Zahlworlen bei den alien Agyptern (Strassburg, 1916), p. 24-29. 3 J. E. Quibell and F. W. Green, Hierakonopolis (London, 1900-1902), Part I, Plate 26B, who describe the victory monument of King Ncr-mr; the number of prisoners taken is given as 120,000, while 400,000 head of cattle and 1,422,000 goats were captured. 4 The evolution of the hieratic writing from the hieroglyphic is explained in G. Moller, Hieratische Palaographie, Vol. I, Nos. 614 ff. The demotic writing 12 A HISTORY OF MATHEMATICAL NOTATIONS 17. The hieroglyphic symbols were I for 1, O for 10, C for 100, I for 1,000, | for 10,000, ^ for 100,000, $ for 1,000,000, Q for 10,000,000. The symbol for 1 represents a vertical staff; that for 1,000 a lotus plant; that for 10,000 a pointing finger; that for 100,000 a burbot; that for 1,000,000 a man in astonishment, or, as more recent Etner Zehner HunJerte TctusettJe n I I 60 M nn A II 100 01 no n ODD nnnn JiL Ann KS not 8100 AAAA a i i rwvi nnn II? CO FIG. 5. Egyptian numerals. Hieroglyphic, hieratic, and demotic numeral symbols. (This table was compiled by Kurt Sethc.) Egyptologists claim, the picture of the cosmic deity Hh. 1 The sym- bols for 1 and 10 are sometimes found in a horizontal position. 18. We reproduce in Figures 5 and 6 two tables prepared by Kurt is explained by F. L. Griffith, Catalogue of the Demotic Papyri in the John Rylands Library (Manchester, 1909), Vol. Ill, p. 415 if., and by H. Brugsch, Grammaire d&motique, 131 ff. 1 Sethe, op.cit., p. 11, 12. OLD NUMERAL SYMBOLS 13 Sethe. They show the most common of the great variety of forms which are found in the expositions given by Moller, Griffith, and Brugsch. Observe that the old hieratic symbol for % was the cross X, sig- nifying perhaps a part obtainable from two sections of a body through the center. Attaeyyfttiscke BruchiticLcn, ftrc&ttckt fruchzetJu* tit in M> IP X Mil mi III! ** lift %'/* tv %'At y< [llllllj nun mm //* *; -90* FIG. 6. Egyptian symbolism for simple fractions. (Compiled by Kurt Sethe) 19. In writing numbers, the Egyptians used the principles of addi- tion and multiplication. In applying the additive principle, not more than four symbols of the same kind were placed in any one group. Thus, 4 was written in hieroglyphs 1 1 1 1 ; 5 was not written HIM, but either 1 1 1 1 1 or , , . There is here recognized the same need which caused the Romans to write V after IIII, L = 50 after XXXX = 40, D = 500 after CCCC = 400. In case of two unequal groups, the Egyp- tians always wrote the larger group before, or above the smaller group; thus, seven was written ,, , . 14 A HISTORY OF MATHEMATICAL NOTATIONS 20. In the older hieroglyphs 2,000 or 3,000 was represented by two or three lotus plants grown in one bush. For example, 2,000 was ^ ; correspondingly, 7,000 was designated by 23K ? . The later hiero- glyphs simply place two lotus plants together, to represent 2,000, with- out the appearance of springing from one and the same bush. 21. The multiplicative principle is not so old as the additive; it came into use about 1600-2000 B.C. In the oldest example hitherto known, 1 the symbols for 120, placed before a lotus plant, signify 120,000. A smaller number written before or below or above a sym- bol representing a larger unit designated multiplication of the larger by the smaller. Mollcr cites a case where 2,800,000 is represented by one burbot, with characters placed beneath it which stand for 28. 22. In hieroglyphic writing, unit fractions were indicated by placing the symbol <o over the number representing the denomina- tor. Exceptions to this arc the modes of writing the fractions | and f ; the old hieroglyph for \ was ^=T, the later was /" ~; of the slightly varying hieroglyphic forms for -|, was quite common. 2 23. We reproduce an algebraic example in hieratic symbols, as it occurs in the most important mathematical document of antiquity known at the present time the Rhind papyrus. The scribe, Ahmcs, who copied this papyrus from an older document, used black and red ink, the red in the titles of the individual problems and in writing auxiliary numbers appearing in the computations. The example which, in the Eisenlohr edition of this papyrus, is numbered 34, is hereby shown. 3 Hieratic writing was from right to left. To facilitate the study of the problem, we write our translation from right to left and in the same relative positions of its parts as in the papyrus, except that numbers are written in the order familiar to us; i.e., 37 is written in our translation 37, and not 73 as in the papyrus. Ahmes writes unit fractions by placing a dot over the denominator, except in case of 1 Ibid., p. 8. 2 Ibid., p. 92-97, gives detailed information on the forms representing f. The Egyptian procedure for decomposing a quotient into unit fractions is explained by V. V. Bobynin in Abh. Gesch. Math., Vol. IX (1899), p. 3. 8 Ein matkematisches Handbook der alien Agypter (Papyrus Rhind des British Museum) t'ibersetzt und erkldrt (Leipzig, 1877; 2d cd., 1891). The explanation of Problem 34 is given on p. 55, the translation on p. 213, the facsimile reproduction on Plate XIII of the first edition. The second edition was brought out without the plates. A more recent edition of the Ahmes papyrus is due to T. Eric Peet and appears under the title The Rhind Mathematical Papyrus, British Museum, Nos. 10057 and 10058, Introduction, Transcription, and Commentary (London, 1923). OLD NUMERAL SYMBOLS 15 i> i> > i> eacn f which had its own symbol. Some of the numeral symbols in Ahmes deviate somewhat from the forms given in the two preceding tables; other symbols are not given in those tables. For the reading of the example in question we give here the following symbols : Four One-fourth X Five "1 Heap S$t See Fig. 7 Seven Q- The whole |J See Fig. 7 One-half ~7 It gives & See Fig. 7 FIG. 7. An algebraic equation and its solution in the Ahmes papyrus, 1700 B.C., or, according to recent authorities, 1550 B.C. (Problem 34, Plate XIII in Eisenlohr; p. 70 in Peet; in chancellor Chace's forthcoming edition, p. 76, as R. C. Archibald informs the writer.) Translation (reading from right to left) : "10 gives it, whole its, \ its, \ its, Heap No. 34 a l 4 j I 5 is heap the together 7 4 1 I Proof the of Beginning -iV 1 1 5 I \- Remainder - 9 together A izV i i 1 I 14 gives i A -^V & TT A I 21 Together .7 gives i 122448" 16 A HISTORY OF MATHEMATICAL NOTATIONS 24. Explanation: oc or The algebraic equation is 0+4+2= 10 i.e., (l+i+i)*=10 The solution answers the question, By what must (1 ^ |) be multiplied to yield the product 10? The four lines 2-5 contain on the right the following computation : Twice (1 H) yields 3 fc. Four times (1 - \) yields 7. One-seventh of (1 \ -J-) is \. t UNITES. . S1GNES LETTRES MIMEIULES copies. VALEUR ik-s S10NES. NOMS OK A01IBBK <m dialecle tliebaiu. UlKROCLYriUQUES , creux ct plciiis. UI8AAT1Q1IBS , uvcc variantcs. i ') I ? ? t Vtl. 00 H H 'M & rt snau. DOD HI ^ 04 TT :t chonwnt. DDOD i! UH -u^ 4 ^ A ftoou. ODD 00 '" 1 1 1 E r> lion. DOD DOD !!! t Z r (> soon. 0000000 V" ^t xti ^&i t. 7 sachf. flDDOOODO m'i ^=^ =* F 8 chinoun. 000 ODD ODD "Si!" ^.^ tf <) pxix. (Continued on facing page] [i.e., taking (1 \ -J-) once, then four times, together with \ of it, yields only 9; there is lacking 1. The remaining computation is on the four lines 2-5, on the left. Since \ of (1 | {) yields (\ -^ ^ 8 ) or -J, lor] (i A) of (1 i 1), yields |. And the double of this, namely, (^ - f ) of (1 \ |) yields 1. Adding together 1, 4, | and (fc iV), we obtain Heap = 5^ ^ -^ or 5f , the answer. OLD NUMERAL SYMBOLS 17 Proof. 5 ^ \ i l f is multiplied by (1 ^ J) and the partial products are added. In the first line of the proof we have 5 ^ | ^ l f , in the second line half of it, in the third line one-fourth of it. Adding at first only the integers of the three partial products and the simpler fractions i> i> !> i> i> the partial sum is 9 -\ \. This is \ I short of 10. In the fourth line of the proof (1. 9) the scribe writes the remaining fractions and, reducing them to the common denominator 56, he writes (in 2 DIZA1NES. SIGiNES LETTRES NUMKRALES copies. VALEUR dcs 8IONK8. NOMS DE NOMDRB en dialcctc thebaih. II1EROGLYPI1IQURS . crcux el plein. IIIERATIQUES , nvcc varianles. X X /6 I 10 ment. ChinVe connmm dos dizaincs : XX * 30 sjouoL maab. mi n ^- -*- H /JO hme. \ *\ 1? 5o taioii. Jii &. I (>o se. x x o" 7 chfe. >ui4 jiU TT 80 hmeiie. ^ Ci 9 ptsldtou. FIG. 8. Hieroglyphic, hieratic, and Coptic numerals. (Taken from A. P. Pihan, Expos6 des signes de numeration [Paris, 1860], p. 26, 27.) red color) in the last line the numerators 8, 4, 4, 2, 2, 1 of the reduced fractions. Their sum is 21. But e w = 5o ^-=-7 o , which is the exact oo 4 8 amount needed to make the total product 10. A pair of legs symbolizing addition and subtraction, as found in impaired form in the Ahmes papyrus, are explained in 200. 25. The Egyptian Coptic numerals are shown in Figure 8. They are of comparatively recent date. The hieroglyphic and hieratic are 18 A HISTORY OF MATHEMATICAL NOTATIONS the oldest Egyptian writing; the demotic appeared later. The Cop- tic writing is derived from the Greek and demotic writing, and was used by Christians in Egypt after the third century. The Coptic numeral symbols were adopted by the Mohammedans in Egypt after their conquest of that country. 26. At the present time two examples of the old Egyptian solu- tion of problems involving what we now term "quadratic equations" 1 are known. For square root the symbol Ir 3 has been used in the modern hieroglyphic transcription, as the interpretation of writing in the two papyri; for quotient was used the symbol oo . PHOENICIANS AND SYRIANS 27. The Phoenicians 2 represented the numbers 1-9 by the re- spective number of vertical strokes. Ten was usually designated by a horizontal bar. The numbers 1 1-19 were expressed by the juxtaposi- tion of a horizontal stroke and the required number of vertical ones. Palmyreaische ZaMzeiebn I X 3 ; 3, JD'p^ ; ,55" 7 . '7^3 3 '''CO" Virianten >ei Oruter / -V ; >. ; , V ; >V ( >.V ''^0><?'>V' BtdeuUag 1. 0; 10. 20 100, 110. 1000 JW. FIG. 9. Palmyra (Syria) numerals. (From M. Cantor, Kulturleben, etc., Fig. 48) As Phoenician writing proceeded from right to left, the horizontal stroke signifying 10 was placed farthest to the right. Twenty was represented by two parallel strokes, either horizontal or inclined and sometimes connected by a cross-line as in H, or sometimes by two strokes, thus A- One hundred was written thus |<| or thus | >| . Phoe- nician inscriptions from which these symbols are taken reach back several centuries before Christ. Symbols found in Palmyra (modern Tadmor in Syria) in the first 250 years of our era resemble somewhat the numerals below 100 just described. New in the Palmyra numer- 1 See H. Schack-Schackcnburg, "Der Berliner Papyrus 6619," Zeitschrift fur dgyptische Sprache und Altertumskunde, Vol. XXXVIII (1900), p. 136, 138, and Vol. XL (1902), p. 6S-66. 2 Our account is taken from Moritz Cantor, Vorlesungen fiber Geschichte der Mathematik, Vol. I (3d ed. ; Leipzig, 1907), p. 123, 124; Mathematische Beitrage zum KuUurleben der Volker (Halle, 1863), p. 255, 256, and Figs. 48 and 49. OLD NUMERAL SYMBOLS 19 als is 7 for 5. Beginning with 100 the Palmyra numerals contain new forms. Placing a I to the right of the sign for 10 (see Fig. 9) signifies multiplication of 10 by 10, giving 100. Two vertical strokes 1 1 mean 10X20, or 200; three of them, 10X30, or 300. 28. Related to the Phoenician are numerals of Syria, found in manuscripts of the sixth and seventh centuries A.D. Their shapes and their mode of combination are shown in Figure 10. The Syrians em- ployed also the twenty-two letters of their alphabet to represent the numbers 1-9, the tens 10-90, the hundreds 100-400. The following hundreds were indicated by juxtaposition: 500 = 400+100, 600 = 400+200, ____ , 900=400+400+100, or else by writing respectively 50-90 and placing a dot over the letter to express that its value is to be taken tenfold. Thousands were indicated by the letters for 1-9, with a stroke annexed as a subscript. Ten thousands were expressed I, H - 2, HI - 3, FP- 4,. -*-5. h-* -6 7 HM-8. H^-9 7-io 7-u K7-12 w, HH^ -18, O - 20 70 - M. TI - 100 Syrische Zahlzeiche.n FIG. 10. Syrian numerals. (From M. Cantor, Kulturleben, etc., Fig. 49) by drawing a small dash below the letters for one's and ten's. Millions were marked by the letters 1-9 with two strokes annexed as sub- scripts (i.e., 1,000X1,000 = 1,000,000). HEBREWS 29. The Hebrews used their alphabet of twenty-two letters for the designation of numbers, on the decimal plan, up to 400. Figure 11 shows three forms of characters: the Samaritan, Hebrew, and Rabbinic or cursive. The Rabbinic was used by commentators of the Sacred Writings. In the Hebrew forms, at first, the hundreds from 500 to 800 were represented by juxtaposition of the sign for 400 and a second number sign. Thus, pn stood for 500, ^n for 600, ISO for 700, nn for 800. 30. Later the end forms of five letters of the Hebrew alphabet came to be used to represent the hundreds 500-900. The five letters representing 20, 40, 50, 80, 90, respectively, had two forms; one of 20 A HISTORY OF MATHEMATICAL NOTATIONS LETTRES MOMS NOMS 8AMAIUTAIRES HEBBAIQUES. BABBI1UQUBS. DBS LKTTRB3. DE NOMBBE. * N (S aleph , a t ekhdd. a 2 5 bet, b 2 chewing 1 : J ghimcl , gh 3 clielochdh. 1 1 1 dalet, d A arbd'ah. * n p y, A 5 khamichdh. * i ) waw, w 6 chichdh. 1 T t zain, z 7 chib'dh. * n p khel, B 8 chemondh. * D V t'et', t' 9 tich'dh. m ' 5 iod, t 10 'asdrdh. a 3 o kaph , k 30 'esrim. 4 h 1) lamed , / do chelochim* D p mem , m %0 arbd'im. A J D noun y n 5o khamichim. * D D s'amek i 60 chichim. v y J> c ain , ' 70 chib'im. 3 s D ph^, ph 80 chemonim. Ytt 3J 3 Lsade, to 9<> tictiim. 1? p P qopli, ^ 100 mtdh. ^ *] 5 rech, r 900 mdtai'm. JJJL t^ ft chin , ch 3oo cluilvchmttt. A n P lau, W400 arba* mcdt. FIG. 11. Hebrew numerals. (Taken from A. P. Pihan, Expos6 des signes de numeration [Paris, 1860], p. 172, 173.) OLD NUMERAL SYMBOLS 21 the forms occurred when the letter was a terminal letter of a word. These end forms were used as follows: Y T| T D 1 900 800 700 600 500. To represent thousands the Hebrews went back to the beginning of their alphabet and placed two dots over each letter. Thereby its value was magnified a thousand fold. Accordingly, represented 1,000. Thus any number less than a million could be represented by their system. 31. As indicated above, the Hebrews wrote from right to left. Hence, in writing numbers, the numeral of highest value appeared on the right; )$n meant 5,001, n& meant 1,005. But 1,005 could be written also flK , where the two dots were omitted, for when ^ meant unity, it was always placed to the left of another numeral. Hence when appearing on the right it was interpreted as meaning 1,000. With a similar understanding for other signs, one observes here the beginning of an imperfect application in Hebrew notation of the principle of local value. By about the eighth century A.D., one finds that the signs iTD^n signify 5,845, the number of verses in the laws as given in the Masora. Here the sign on the extreme right means 5,000; the next to the left is an 8 and must stand for a value less than 5,000, yet greater than the third sign representing 40. Hence the sign for 8 is taken here as 800. l GREEKS 32. On the island of Crete, near Greece, there developed, under Egyptian influence, a remarkable civilization. Hieroglyphic writing on clay, of perhaps about 1500 B.C., discloses number symbols as follows: ) or I for 1, ))))) or 1 1 1 1 1 or "' for 5, for 10, \ or / for 100, <> for 1,000, V for i (probably), \\\\: :::))) for 483. 2 In thk combination of symbols only the additive principle is employed. Somewhat later, 3 10 is represented also by a horizontal dash; the 1 G. H. F. Ncsselmann, Die Algebra der Griechen (Berlin, 1842), p. 72, 494; M. Cantor, Vorlesungen liber Geschichte der Malhematik, Vol. I (3d ed.), P- 126, 127. 2 Arthur J. Evans, Scripla Minoa, Vol. I (1909), p. 258, 256. 8 Arthur J. Evans, The Palace of Minos (London, 1921), Vol. 1, p 646; see also p. 279. 22 A HISTORY OF MATHEMATICAL NOTATIONS sloping line indicative of 100 and the lozenge-shaped figure used for 1,000 were replaced by the forms O for 100, and <> for 1,000. OOo o = = = I I I stood for 2 > 496 33. The oldest strictly Greek numeral symbols were the so-called Hcrodianic signs, named after Herodianus, a Byzantine grammarian of about 200 A.D., who describes them. These signs occur frequently in Athenian inscriptions and are, on that account, now generally called Attic. They were the initial letters of numeral adjectives. 1 They were used as early as the time of Solon, about GOO B.C., and con- tinued in use for several centuries, traces of them being found as late as the time of Cicero. From about 470 to 350 B.C. this system existed in competition with a newer one to be described presently. The Herodianic signs were 1 Iota for 1 II Eta for 100 II or T I or F Pi for 5 X Chi for 1,000 A Delta for 10 M My for 10,000 34. Combinations of the symbols for 5 with the symbols for 10,100, 1,000 yielded symbols for 50, 500, 5,000. These signs appear on an abacus found in 1847, represented upon a Greek marble monument on the island of Salamis. 2 This computing table is represented in Fig- ure 12. The four right-hand signs I C T X, appearing on the horizontal line below, stand for the fractions -J, ^, -, 4 * 8 , respectively. Proceed- ing next from right to left, we have the symbols for 1, 5, 10, 50, 100, 500, 1,000, 5,000, and finally the sign T for 6,000. The group of sym- bols drawn on the left margin, and that drawn above, do not contain the two symbols for 5,000 and 6,000. The pebbles in the columns represent the number 9,823. The four columns represented by the five vertical lines on the right were used for the representation of the fractional values J, -^-5-, ^ J 4 , 4 J, respectively. 35. Figure 13 shows the old Herodianic numerals in an Athenian state record of the fifth century B.C. The last two lines are: Ke0<xA(uoj> 1 See, for instance. G. Friedlein, Die Zahlzeichen und das elementarc Rechnen dcr Griechen und Romer (Krlangen, 1869), p. 8; M. Cantor, Vorlesungen uber Geschichte der Mathemaiik, Vol. I (3d ed.), p. 120; II. Ilankel, Zur Geschichte der Mathemalik im Alter thum und Mittelalter (Leipzig, 1874), p. 37. 2 Kubitschek, "Die Salaminische Rechentafel," Numismatische Zeitschrift (Vienna, 1900), Vol. XXXI, p. 393-98; A. Nagl, ibid., Vol. XXXV (1903), p. 131- 43; M. Cantor, Kulturleben der Volker (Halle, 1863), p. 132, 136; M. Cantor, Vor- lesungen uber Geschichte der Mathematik, Vol. I (3d ed.), p. 133. OLD NUMERAL SYMBOLS 23 ai/a[Xcoarosr] oD eiri r[r?s] apxw HHHPTTT....; i.e., "Total of expenditures during our office three hundred and fifty-three talents " 36. The exact reason for the displacement of the Herodianic sym- bols by others is not known. It has been suggested that the com- mercial intercourse of Greeks with the Phoenicians, Syrians, and Hebrews brought about the change. The Phoenicians made one im- portant contribution to civilization by their invention of the alpha- bet. The Babylonians and Egyptians had used their symbols to represent whole syllables or words. The Phoenicians borrowed hieratic X o JL < a. E. x TPXPHPAPHCTX FIG. 12. The computing table of Salamis signs from Egypt and assigned them a more primitive function as letters. But the Phoenicians did not use their alphabet for numerical purposes. As previously seen, they represented numbers by vertical and horizontal bars. The earliest use of an entire alphabet for desig- nating numbers has been attributed to the Hebrews. As previously noted, the Syrians had an alphabet representing numbers. The Greeks are supposed by some to have copied the idea from the He- brews. But Moritz Cantor 1 argues that the Greek use is the older and that the invention of alphabetic numerals must be ascribed to the Greeks. They used the twenty-four letters of their alphabet, together with three strange and antique letters, ST (old van), 9 (koppa), *) (sampi), and the symbol M. This change was decidedly for the worse, for the old Attic numerals were less burdensome on the memory inas- 1 V&rlesungen uber Geschichte der Mathematik, Vol. I (3d ed., 1907), p. 25. 24 A HISTORY OF MATHEMATICAL NOTATIONS FIG. 13. Account of disbursements of the Athenian state, 418-415 B.C., British Museum, Greek Inscription No. 23. (Taken from R. Brown, A History of Accounting and Accountants [Edinburgh, 1905], p. 26.) OLD NUMERAL SYMBOLS 25 much as they contained fewer symbols. The following are the Greek alphabetic numerals and their respective values: aftyde^frjBi K X/i v o TT 9 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 P <r r v <p x t ^ ,a ,/3 ,7, 100 200 300 400 500 600 700 800 900 1,000 2,000 3,000 etc. P v M M M, etc. 10,000 20,000 30,000 37. A horizontal line drawn over a number served to distinguish it more readily from words. The coefficient for M was sometimes placed before or behind instead of over the M. Thus 43,678 was written SM^x^- The horizontal line over the Greek numerals can hardly be considered an essential part of the notation ; it does not seem to have been used except in manuscripts of the Byzantine period. 1 For 10,000 or myriad one finds frequently the symbol M or Mu, sometimes simply the dot , as in /3-o5 for 20,074. Often 2 the coefficient of the myriad is found written above the symbol /i u . 38. The paradox recurs, Why did the Greeks change from the Herodianic to the alphabet number system? Such a change would not be made if the new did not seem to offer some advantages over the old. And, indeed, in the new system numbers could be written in a more compact form. The Herodianic representation of 1,739 was X HlHIIAAAII MM; the alphabetic was ,a^X0. A scribe might consider the latter a great innovation. The computer derived little aid from either. Some advantage lay, however, on the side of the Herodianic, as Cantor pointed out. Consider HHIIH+HH= SI H,AAAA+AA = S|A; there is an analogy here in the addition of hundred's and of ten's. But no such analogy presents itself in the alphabetic numerals, where the corresponding steps are v+a = x and /Z+K = ; adding the hun- dred's expressed in the newer notation affords no clew as to the sum of the corresponding ten's. But there was another still more impor- tant consideration which placed the Herodianic far above the alpha- betical numerals. The former had only six symbols, yet they afforded an easy representation of numbers below 100,000; the latter demanded twenty-seven symbols for numbers below 1,000! The mental effort 1 Encyc. des stien. math., Tome I, Vol. I (1904), p. 12. 2 Ibid. 26 A HISTORY OF MATHEMATICAL NOTATIONS of remembering such an array of signs was comparatively great. We are reminded of the centipede having so many legs that it could hardly advance. 39. We have here an instructive illustration of the fact that a mathematical topic may have an amount of symbolism that is a hin- drance rather than a help, that becomes burdensome, that obstructs progress. We have here an early exhibition of the truth that the move- ments of science are not always in a forward direction. Had the Greeks not possessed an abacus and a finger symbolism, by the aid of which computations could be carried out independently of the numeral notation in vogue, their accomplishment in arithmetic and algebra might have been less than it actually was. 40. Notwithstanding the defects of the Greek system of numeral notation, its use is occasionally encountered long after far better systems were generally known. A Calabrian monk by the name of Barlaam, 1 of the early part of the fourteenth century, wrote several mathematical books in Greek, including arithmetical proofs of the second book of Euclid's Elements, and six books of Logistic, printed in 1564 at Strassburg arid in several later editions. In the Logistic he de- velops the computation with integers, ordinary fractions, and sexa- gesimal fractions; numbers are expressed by Greek letters. The appearance of an arithmetical book using the Greek numerals at as late a period as the close of the sixteenth century in the cities of Strass- burg and Paris is indeed surprising. 41. Greek writers often express fractional values in words. Thus Archimedes says that the length of a circle amounts to three diameters and a part of one, the size of which lies between one-seventh and ten- seventy-firsts. 2 Eratosthenes expresses J J of a unit arc of the earth's meridian by stating that the distance in question "amounts to eleven parts of which the meridian has eighty-three. " 3 When expressed in symbols, fractions were often denoted by first writing the numerator marked with an accent, then the denominator marked with two ac- cents and written twice. Thus, 4 if KCL" KCL" = |f. Archimedes, Euto- cius, and Diophantus place the denominator in the position of the 1 All our information on Barlaam is drawn from M. Cantor, Vorlesungen liber Geschichte der Matkematik, Vol. I (3d ed.), p. 509, 510; A. G. Kastner, Geschichte der Mathematik (Gottingen, 1796), Vol. I, p. 45; J. C. Hcilbronner, Historia matheseos universae (Lipsiae, 1742), p. 488, 489. 2 Archimedis opera omnia (ed. Heiberg; Leipzig, 1880), Vol. I, p. 262. 3 Ptolemaus, MeyaXij avvrafa (ed. Heiberg), Pars I, Lib. 1, Cap. 12, p. 68. 4 Heron, Stereometrica (ed. Hultsch; Berlin, 1864), Pars I, Par. 8, p. 155. OLD NUMERAL SYMBOLS 27 modern exponent; thus 1 Archimedes and Eutocius use the notation __ KO! Ka if or if for ^], and Diophantus ( 101-6), in expressing large num- bers, writes (Ariihmetica, Vol. IV, p. 17), ^ for -w-L.-,- . 7-/^X/ca 2,704 Here the sign ~ takes the place of the accent. Greek writers, even as late as the Middle Ages, display a preference for unit fractions, which played a dominating role in old Egyptian arithmetic. 2 In expressing such fractions, the Greeks omitted the a for the numerator and wrote the denominator only once. Thus ju6 //= 4 V- Unit fractions in juxta- position were added, 3 as in f" /cr?" pt/3" o-/c6 // = ^+^V+iH+ T*4- ^ ne finds also a single accent, 4 as in 5' = \. Frequent use of unit fractions is found in Gcminus (first century B.C.), Diophantus (third century A.D.), Eutocius and Proclus (fifth century A.D.). The fraction \ had a mark of its own, 5 namely, L or , but this designation was no more adopted generally among the Greeks than were the other notations of fractions. Ptolemy 6 wrote 3850' (i.e., 38 ,i |) thus, XT;' '7'". Hultsch has found in manuscripts other symbols for |, namely, the semicircles VI , (, and the sign ,S ; the origin of the latter is uncertain. He found also a symbol for , resembling somewhat the small omega (co). 7 Whether these symbols represent late practice, but not early usage, it is difficult to determine with certainty. 42. A table for reducing certain ordinary fractions to the sum of unit fractions is found in a Greek papyrus from Egypt, described by 1 G. II. F. Nessclmarm, Algebra der Gricchcn (Berlin, 1842), p. 114. 2 ,1. Baillct describes a papyrus, "Le papyrus mathematique d'Akhmfm," in Memoires publics par Ics immbrcs de la Mission archeologique fran^aise au Caire (Paris, 1892), Vol. IX, p. 1-89 (8 plates). This papyrus, found at Akhmtrn, in Egypt, is written in Greek, and is supposed to belong to the period between 500 and 800 A.D. It contains a table for the conversion of ordinary fractions into unit frac- tions. 3 Fr. Hultsch, Metrologicorum scriplorum reliquiae (1864-66), p. 173-75; M. Cantor, Vorlesungen iiber Geschichte der Mathematik, Vol. I (3d ed.), p. 129. 4 Nesselmann, op. cil., p. 112. 5 Ibid.; James Gow, Short History of Greek Mathematics (Cambridge, 1884), p. 48, 50. *Geographia (ed. Carolus Mullerus; Paris, 1883), Vol. I, Part I, p. 151. 7 Metrologicorum scriptorum reliquiae (Leipzig, 1804), Vol. I, p. 173, 174. On p. 175 and 176 Hultsch collects the numeral symbols found in three Parisian manu- scripts, written in Greek, which exhibit minute variations in the symbolism. For instance, 700 is found to be ^ ^, \j/ f . 28 A HISTORY OF MATHEMATICAL NOTATIONS L. C. Karpinski, 1 and supposed to be intermediate between the Ahmes papyrus and the Akhmim papyrus. Karpinski (p. 22) says: "In the table no distinction is made between integers and the corre- sponding unit fractions; thus 7' may represent either 3 or , and actually y'y' in the table represents 3^. Commonly the letters used as numerals were distinguished in early Greek manuscripts by a bar placed above the letters but not in this manuscript nor in the Akhmim papyrus. 7 ' In a third document dealing with unit fractions, a Byzan- tine table of fractions, described by Herbert Thompson, 2 f is written 1; i, a; J, f (from \ '); |, A/" (from A'); *, e (from e'); i, vf (from H'). As late as the fourteenth century, Nicolas Rhabdas of Smyrna wrote two letters in the Greek language, on arithmetic, containing tables for unit fractions. 3 Here letters of the Greek alphabet used as integral numbers have bars placed above them. 43. About the second century before Christ the Babylonian sexa- gesimal numbers were in use in Greek astronomy; the letter omicron, which closely resembles in form our modern zero, was used to desig- nate a vacant space in the writing of numbers. The Byzantines wrote it usually b, the bar indicating a numeral significance as it has when placed over the ordinary Greek letters used as numerals. 4 44. The division of the circle into 360 equal parts is found in Hypsicles. 5 Hipparchus employed sexagesimal fractions regularly, as did also C. Ptolemy 6 who, in his Almagest, took the approximate 8 30 value of TT to be 3+^+^-^ ^ n ^ ne Heiberg edition this value is written 7 rj X, purely a notation of position. In the tables, as printed by Heiberg, the dash over the letters expressing numbers is omitted. In the edition of N. Halma 7 is given the notation 7 ?/ X", which is 1 "The Michigan Mathematical Papyrus No. 621," Isis, Vol. V (1922), p. 20-25. 2 "A Byzantine Table of Fractions," Ancient Egypt, Vol. I (1914), p. 52-54. 3 The letters were edited by Paul Tannery in Notices et extraits des manuscrits de la Bibliotheque Nationale, Vol. XXXII, Part 1 (1886), p. 121-252. 4 C. Ptolemy, Almagest (ed. N. Halma; Paris, 1813), Book I, chap, ix, p. 38 and later; J. L. Heiberg, in his edition of the Almagest (Syntaxis mathematical (Leipzig, 1898; 2d ed., Leipzig, 1903), Book I, does not write the bar over the o but places it over all the significant Greek numerals. This procedure has the ad- vantage of distinguishing between the o which stands for 70 and the o which stands for zero. See Encyc. des scien. math., Tome I, Vol. I (1904), p. 17, n. 89. 5 Ava<optKos (ed. K. Manitius), p. xxvi. 6 Syntaxis mathematica (ed. Heiberg), Vol. I, Part 1, p. 513. 7 Composition math, de PtoUmee (Paris, 1813), Vol. I, p. 421; see also Encyc. des scien. math., Tome I, Vol. I (1904), p. 53, n. 181. OLD NUMERAL SYMBOLS 29 probably the older form. Sexagesimal fractions were used during the whole of the Middle Ages in India, and in Arabic and Christian coun- tries. One encounters them again in the sixteenth and seventeenth centuries. Not only sexagesimal fractions, but also the sexagesimal notation of integers, are explained by John Wallis in his Maihesis universalis (Oxford, 1657), page 68, and by V. Wing in his Astronomia Briiannica (London, 1652, 1669), Book I. EARLY ARABS 45. At the time of Mohammed the Arabs had a script which did not differ materially from that of later centuries. The letters of the early Arabic alphabet came to be used as numerals among the Arabs 1 t 5 6 20 : so <! 40 r 50 c 60 70 80 90 ^ 100 d .200 ^ 300^ 400 c 500 & 600 700 o 800 o 900 Jb 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 j, 20000 J 30000 jj 40000 o 50000 jj 60000 ^ 70000 .* 80000 90000 100000 g 200000 , 300000 jxi 400000 43 500000 J3 600000 j> 700000 ^ 800000 -* i 900000 FIG. 14. Arabic alphabetic numerals used before the introduction of the Hindu-Arabic numerals. as early as the sixth century of our era. 1 After the time of Mohammed, the conquering Moslem armies coming in contact with Greek culture acquired the Greek numerals. Administrators and military leaders used them. A tax record of the eighth century contains numbers expressed by Arabic letters and also by Greek letters. 2 Figure 14 is a table given by Ruska, exhibiting the Arabic letters and the numerical values which they represent. Taking the symbol for 1,000 twice, on the multiplicative principle, yielded 1,000,000. The Hindu-Arabic 1 Julius Ruska, "Zur altesten arabischen Algebra und Rechenkunst," Sitzungs- berichte d. Heidelberger Akademie der Wissensch. (Philos.-histor. Klasse, 1917; 2. Abhandlung), p. 37. 2 Ibid., p. 40. 30 A HISTORY OF MATHEMATICAL NOTATIONS numerals, with the zero, began to spread among the Arabs in the nin and tenth centuries, and they slowly displaced the Arabic and Gre< numerals. 1 ROMANS 46. We possess little definite information on the origin of tl Roman notation of numbers. The Romans never used the successr letters of their alphabet for numeral purposes in the manner practic< by the Syrians, Hebrews, and Greeks, although (as we shall see) * alphabet system was at one time proposed by a late Roman write Before the ascendancy of Rome the Etruscans, who inhabited ti country nearly corresponding to modern Tuscany and who ruled Rome until about 500 B.C., used numeral signs which resembled lette of their alphabet and also resembled the numeral signs used by tl Romans. Moritz Cantor 2 gives the Etrurian and the old Roman sign as follows: For 5, the Etrurian /\ or V, the old Roman V; for 10 tl Etrurian X or +, the old Roman X; for 50 the Etrurian t or I, tl old Roman "f or I or X or 1 or L; for 100 the Etrurian 0, the o Roman ; for 1,000 the Etrurian #, the old Roman 0. The reser blance of the Etrurian numerals to Etrurian letters of the alphabet seen from the following letters: V, +, I, O, 8. These resemblanc cannot be pronounced accidental. "Accidental, on the other hand says Cantor, "appears the relationship with the later Roman signs, V, X, L, C, M, which from their resemblance to letters transformc themselves by popular etymology into these very letters/ 7 The origii of the Roman symbols for 100 and 1,000 are uncertain; those for ' and 500 are generally admitted to be the result of a bisection of tl two former. "There was close at hand/' says G. Friedlein, 3 "the a breviation of the word centum and mille which at an early age brougl about for 100 the sign C, and for 1,000 the sign M and after Augustu M." A view held by some Latinists 6 is that "the signs for 50, 10 1,000 were originally the three Greek aspirate letters which the R mans did not require, viz., M>, O, 0, i.e., x> 0> * The *& was writte J_ and abbreviated into L; O from a false notion of its origin made HI 1 Ibid., p. 47. 2 Vorlesungen uber Geschichte der Mathematik, Vol. I (3d ed.), p. 523, and t' table at the end of the volume. 3 Die Zahlzeichen und das elementare Rechnen der Griechen und Homer (E langen, 1869), p. 28. 4 Theodor Mommsen, Die unleritalischen Dialekte (Leipzig, 1840), p. 30. 'Ritschl, Rhein. Mus., Vol. XXIV (1869), p. 12. OLD NUMERAL SYMBOLS 31 the initial of centum; and assimilated to ordinary letters CIO. The half of 0, viz., D, was taken to be -J- 1,000, i.e., 500; X probably from the ancient form of 0, viz., , being adopted for 10, the half of it V was taken for 5." 1 47. Our lack of positive information on the origin and early his- tory of the Roman numerals is not due to a failure to advance working hypotheses. In fact, the imagination of historians has been unusually active in this field. 2 The dominating feature in the Roman notation is the principle of addition, as seen in II, XII, CC, MDC, etc. 48. Conspicuous also is the frequent use of the principle of sub- traction. If a letter is placed before another of greater value, its value is to be subtracted from that of the greater. One sees this in IV, IX, XL. Occasionally one encounters this principle in the Baby- lonian notations. Remarks on the use of it are made by Adriano Cappelli in the following passage : "The well-known rule that a smaller number, placed to the left of a larger, shall be subtracted from the latter, as 0|00 = 4,000, etc., was seldom applied by the old Romans and during the entire Middle Ages one finds only a few instances of it. The cases that I have found belong to the middle of the fifteenth century and are all cases of IX, never of IV, and occurring more especially in French and Piedmontese documents. Walther, in his Lexicon diptomaticum, Gottingen, 1745- 47, finds the notation LXL = 90 in use in the eighth century. On the other hand one finds, conversely, the numbers IIIX, VIX with the meaning of 13 and 16, in order to conserve, as Lupi remarks, the Latin terms tertio dedmo and sexto decimo."* L. C. Karpinski points out that the subtractive principle is found on some early tombstones and on a signboard of 130 B.C., where at the crowded end of a line 83 is written XXCIII, instead of LXXXIII. 1 II. J. Roby, A Grammar of the Latin Language from Plaulus to Suetonius (4th ed.; London, 1881), Vol. I, p. 441. 2 Consult, for example, Friedlcin, op. cit., p. 26-31; Ncsselmann, op. tit., p. 86-92; Cantor, Mathematische Beitrdge zum Kulturleben der Volker, p. 155-67; J. C. Heilbronner, Historia Matheseos universae (Lipsiae, 1742), p. 732-35; Grotc- fend, Lateinische Grammatik (3d ed.; Frankfurt, 1820), Vol. II, p. 163, is quoted in the article "Zahlzeichen" in G. S. Kliigel's Malhematisches Worterbuch, continued by C. B. Mollweide and J. A. Grunert (Leipzig, 1831); Mommsen, Hermes, Vol. XXII (1887), p. 596; Vol. XXIII (1888), p. 152. A recent discussion of the history of the Roman numerals is found in an article by Ettore Bortolotti in Bolletino delta Mathesis (Pavia,. 1918), p. 60-66, which is rich in bibliographical references, as is also an article by David Eugene Smith in Scientia (July- August, 1926). 3 Lexicon Abbreviaturarum (Leipzig, 1901), p. xlix. 32 A HISTORY OF MATHEMATICAL NOTATIONS 49. Alexander von Humboldt 1 makes the following observations: "Summations by juxtaposition one finds everywhere among the Etruscans, Romans, Mexicans and Egyptians; subtraction or lessen- ing forms of speech in Sanskrit among the Indians: in 19 or unavinsati; 99 unusata; among the Romans in undeviginti for 19 (unus de viginti), undeoctoginta for 79; duo de quadraginta for 38; among the Greeks tikosi deonta henos 19, and pentekonta duoin deontoin 48, i.e., 2 missing in 50. This lessening form of speech has passed over in the graphics of numbers when the group signs for 5, 10 and even their multiples, for example, 50 or 100, are placed to the left of the characters they modify (IV and IA, XL and XT for 4 and 40) among the Romans and Etrus- cans (Otfried Miiller, Etrusker, II, 317-20), although among the latter, according to Otfried Miiller's new researches, the numerals descended probably entirely from the alphabet. In rare Roman inscriptions which Marini has collected (Iscrizioni della Villa di Albano, p. 193; Hervas, Aritmetica delle nazioni [1786], p. 11, 16), one finds even 4 units placed before 10, for example, IIIIX for 6." 50. There are also sporadic occurrences in the Roman nota- tions of the principle of multiplication, according to which VM does not stand for 1,000 5, but for 5,000. Thus, in Pliny's His- toria naturalis (about 77 A.D.), VII, 26; XXXIII, 3; IV praef., one finds 2 LXXXIII.M, XCII.M, CX.M for 83,000, 92,000, 110,000, respectively. 51. The thousand-fold value of a number was indicated in some instances by a horizontal line placed above it. Thus, Aelius Lam- pridius (fourth century A.D.) says in one place, "CXX, equitum Persa- rum fudimus: et mox X in bello interemimus," where the numbers designate 120,000 and 10,000. Strokes placed on top and also on the sides indicated hundred thousands; e.g., |X|CLXXXDC stood for 1,180,600. In more recent practice the strokes sometimes occur only on the sides, as in | X | DC . XC . , the date on the title-page of Sigii- enza's Libra astronomicaj published in the city of Mexico in 1690. In antiquity, to prevent fraudulent alterations, XXXM was written for 30,000, and later still CIO took the place of M. 3 According to 1 "liber die bei verschiedenen Volkern ublichen Systeme von Zahlzeichen, etc./' Crclle's Journal fur die reine und angewandte Mathematik (Berlin, 1829), Vol. IV, p. 210, 211. 2 Nesselmann, op. cit., p. 90. 3 Confer, on this point, Theodor Mommsen and J. Marquardt, Manuel des antiquites romaines (trans. G. Humbert), Vol. X by J. Marquardt (trans. A. Vigie"; Paris, 1888), p. 47, 49. OLD NUMERAL SYMBOLS 33 Cappelli 1 "one finds, often in French documents of the Middle Ages, the multiplication of 20 expressed by two small x's which are placed as exponents to the numerals III, VI, VIII, etc., as in IIII XX = 80, VI XX XI = 131." 52. A Spanish writer 2 quotes from a manuscript for the year 1392 the following: M C "IIII, IIII, LXXIII florins" for 4,473 florins. M XX "III C IIII III florins" for 3,183 (?) florins. In a Dutch arithmetic, printed in 1771, one finds 8 c c m c t ffitj for 123, i jttj ittj toj for 123,456. 53. For 1,000 the Romans had not only the symbol M, but also I, oo and CIO. According to Priscian, the celebrated Latin grammarian of about 500 A.D., the oo was the ancient Greek sign X for 1,000, but modified by connecting the sides by curved lines so as to distinguish it from the Roman X for 10. As late as 1593 the oo is used by C. Dasypo- dius 4 the designer of the famous clock in the cathedral at Strasbourg. The CIO was a I inclosed in parentheses (or apostrophes). When only the right-hand parenthesis is written, 10, the value represented is only half, i.e., 500. According to Priscian, 5 "quinque milia per I et duas in dextera parte apostrophes, 100- decem milia per supra dictam formam additis in sinistra parte contrariis duabus notis quam sunt apostrophi, CCIOO." Accordingly, 100 stood for 5,000, CCIOO for 10,000; also 1000 represented 50,000; and CCCIOOO, 100,000; (co), 1,000,000. If we may trust Priscian, the symbols that look like the letters C, or those letters facing in the opposite direction, were not really letters C, but were apostrophes or what we have called 1 Op. cit., p. xlix. 2 Liciniano Saez, Demostracidn Histdrica del verdadero valor de *Todas Las Monedas que corrlan en Castilla durante el reynado del Senor Don Enrique III (Madrid, 1796). 3 De Vernieuwde Cyfferinge van Mf Willem B art j ens. Herstelt, .... door M r Jan van Dam, .... en van alle voorgaande Fauten gezuyvert door .... Klaas Bosch (Amsterdam, 1771), p. 8. 4 Cunradi Dasypodii Institutionum Mathematicarum voluminis primi Erotemata (1593), p. 23. . 6 "De figuris numerorum," Henrici Keilii Grammaiid Latini (Lipsiae, 1859), Vol. Ill, 2, p. 407. 34 A HISTORY OF MATHEMATICAL NOTATIONS parentheses. Through Priscian it is established that this notation is at least as old as 500 A.D. ; probably it was much older, but it was not widely used before the Middle Ages. 54. While the Hindu- Arabic numerals became generally known in Europe about 1275, the Roman numerals continued to hold a com- manding place. For example, the fourteenth-century banking-house of Peruzzi in Florence Compagnia Peruzzi did not use Arabic numerals in their account-books. Roman numerals were used, but the larger amounts, the thousands of lira, were written out in words; one finds, for instance, "Ib. quindicimilia CXV / V ^ VI in fiorini" for 15,115 lira 5 soldi 6 denari; the specification being made that the lira are lira a fiorino cVoro at 20 soldi and 12 denari. There appears also a symbol much like ? , for thousand. 1 Nagl states also: "Specially characteristic is .... during all the Middle Ages, the regular prolongation of the last I in the units, as VI |= VI I, which had no other purpose than to prevent the subsequent addition of a further unit/' 55. In a book by H. Giraua Tarragones 2 at Milan the Roman numerals appear in the running text and are usually underlined; in the title-page, the date has the horizontal line above the numerals. The Roman four is 1 1 1 1 . In the tables, columns of degrees and minutes are headed "G.M."; of hour and minutes, "H.M." In the tables, the Hindu-Arabic numerals appear ; the five is printed 3 , without the usual upper stroke. The vitality of the Roman notation is illustrated further by a German writer, Sebastian Frank, of the sixteenth cen- tury, who uses Roman numerals in numbering the folios of his book and in his statistics: "Zimmet kuinpt von Zailon .CC.VN LX. teiitscher meil von Calicut weyter gelegen ..... Die Nagelin kummen von Meluza / fur Calicut hinaussgelegen vij-c. vnd XL. deutscher meyl." 3 The two numbers given are 260 and 740 German miles. Pe- culiar is the insertion of vnd ("and")- Observe also the use of the principle of multiplication in vijc. ( = 700). In Jakob Kobel's Rechenbiechlin (Augsburg, 1514), fractions appear in Roman numerals; 11 thus, c~ Stands f r *^* 1 Alfred Nagl, Zeitschrift fur Mathematik und Physik, Vol. XXXIV (1889), Historisch-literarische Abthcilung, p. 164. 2 Dos Libros de Cosmographie, complicates nueuamcnte por Hieronymo Giraua Tarragones (Milan, M.D.LVI). 8 Weltbuch I siriegel vnd bildtnis des gantzen Erdtbodens .... von Sebastiano Franco W&rdensi ____ (M.D. XXXIIII), fol. ccxx. OLD NUMERAL SYMBOLS 35 56. In certain sixteenth-century Portuguese manuscripts on navi- gation one finds the small letter b used for 5, and the capital letter R for 40. Thus, z&iij stands for 18, Rii] for 43. 1 to FIG. 15. Degenerate forms of Roman numerals in English archives (Common Pleas, Plea Rolls, 637, 701, and 817; also Recovery Roll 1). (Reduced.) A curious development found in the archives of one or two English courts of the fifteenth and sixteenth centuries 2 was a special Roman 1 J. I. de Brito Rcbcllo, Livro de Marinharia (Lisboa, 1903), p. 37, 85-91, 193, 194. 2 Antiquaries Journal (London, 1926), Vol. VI, p. 273, 274. 36 A HISTORY OF MATHEMATICAL NOTATIONS numeration for the membranes of their Rolls, the numerals assuming a degraded form which in its later stages is practically unreadable. In Figure 15 the first three forms show the number 147 as it was written in the years 1421, 1436, and 1466; the fourth form shows the number 47 as it was written in 1583. 57. At the present time the Roman notation is still widely used in marking the faces of watches and clocks, in marking the dates of books on title-pages, in numbering chapters of books, and on other occasions calling for a double numeration in which confusion might arise from the use of the same set of numerals for both. Often the Roman numerals are employed for aesthetic reasons. 58. A striking feature in Roman arithmetic is the partiality for duodecimal fractions. Why duodecimals and not decimals? We can only guess at the answer. In everyday affairs the division of units into two, three, four, and six equal parts is the commonest, and duodecimal fractions give easier expressions for these parts. Nothing definite is known regarding the time and place or the manner of the origin of these fractions. Unlike the Greeks, the Romans dealt with concrete fractions. The Roman as, originally a copper coin weighing one pound, was divided into 12 unciae. The abstract fraction \-\- was called deuna ( = de unaa, i.e., as [1] less uncia [r 2 ]). Each duodecimal subdivision had its own name and symbol. This is shown in the follow- ing table, taken from Friedlein, 1 in which S stands for semis or "half" of an as. TABLE as 1 n f t I 7 * 5 T2 i 1 1 SK. deunx S r = - or S : : S = = or S : : S = - or S r 1 or & : - or _ _ or : $ or 6 Y< S r or or : : or X S or : : or 1 or : or z or : -LL-Ii't - or or on bronze abacus ( :>ccur also curved ones /^/. (de uncia 1 -fy) f(de sextans 1~J) \ (decem unciae) (de quadrans 1 J) (duae assis sc. partes) (scptem unciae) dextans 1 (decunx)J dodrans bes septunx semis quincunx (quinque unciae) triens . ... quadrans . . sextans . . . sescuncia 1J uncia ^ In place of straight lines - < 1 Op. cit., Plate 2, No. 13; see also p. 35. OLD NUMERAL SYMBOLS 37 59. Not all of these names and signs were used to the same ex- tent. Since i+i=f, there was used in ordinary life | and (semis et triens) in place of $ or \\ (decunx). Nor did the Romans confine them- selves to the duodecimal fractions or their simplified equivalents 1; i> l> 1> etc., but used, for instance, T V in measuring silver, a libella being T V denarius. The uncia was divided in 4 sicilici y and in 24 scripuli etc. 1 In the Geometry of Boethius the Roman symbols are omitted and letters of the alphabet are used to represent fractions. Very probably this part of the book is not due to Boethius, but is an inter- polation by a writer of later date. 60. There are indeed indications that the Romans on rare occa- sions used letters for the expression of integral numbers. 2 Theodor Mommsen and others discovered in manuscripts found in Bern, Einsiedeln, and Vienna instances of numbers denoted by letters. Tartaglia gives in his General trattato di nvmeri, Part I (1556), folios 4, 5, the following: A 500 II R 80 B 300 K 51 S 70 C 100 L 50 T 160 D 500 M 1,000 V 5 E 250 N 90 X 10 F 40 11 Y 150 G 400 P 400 Z 2,000 H 200 Q 500 61. Gerbert (Pope Sylvestre II) and his pupils explained the Ro- man fractions. As reproduced by Olleris, 3 Gerbert's symbol for \ does not resemble the capital letter $, but rather the small letter <J . 1 For additional details and some other symbols used by the Romans, consult Friedlein, p. 33-46 and Plate 3; also H. Hankel, op. tit., p. 57-61, where com- putations with fractions are explained. Consult also Fr. Hultsch, Metrologic. scriplorcs Romani (Leipzig, 1866). 2 Friedlein, op. tit., p. 20, 21, who gives references. In the Standard Dic- tionary of the English Language (New York, 1896), under S, it is stated that 3 stood for 7 or 70. 1 (Euvres de Gerbert (Paris, 1867), p. 343-48, 393-96, 583, 584. 38 A HISTORY OF MATHEMATICAL NOTATIONS PERUVIAN AND NORTH AMERICAN KNOT RECORDS 1 ANCIENT QUIPU 62. 'The use of knots in cords for the purpose of reckoning, and recording numbers" was practiced by the Chinese and some other ancient people; it had a most remarkable development among the Inca of Peru, in South America, who inhabited a territory as large as the United States east of the Rocky Mountains, and were a people of superior mentality. The period of Inca supremacy extended from about the eleventh century A.D. to the time of the Spanish conquest in the sixteenth century. The quipu was a twisted woolen cord, upon which other smaller cords of different colors were tied. The color, length, and number of knots on them and the distance of one from another all had their significance. Specimens of these ancient quipu have been dug from graves. 63. We reproduce from a work by L. Leland Locke a photograph of one of the most highly developed quipu, along with a line diagram of the two right-hand groups of strands. In each group the top strand usually gives the sum of the numbers on the four pendent strands. Thus in the last group, the four hanging strands indicate the numbers 89, 258, 273, 38, respectively. Their sum is 658; it is recorded by the top string. The repetition of units is usually expressed by a long knot formed by tying the overhand knot and passing the cord through the loop of the knot as many times as there are units to be denoted. The numbers were expressed on the decimal plan, but the quipu were not adopted for calculation; pebbles and grains of maize were used in com- puting. 64. Nordenskiold shows that, in Peru, 7 was a magic number; for in some quipu, the sums of numbers on cords of the same color, or the numbers emerging from certain other combinations, are multiples of 7 or yield groups of figures, such as 2777, 777, etc. The quipu dis- close also astronomical knowledge of the Peruvian Indians. 2 65. Dr. Leslie Spier, of the University of Washington, sends me the following facts relating to Indians in North America: "The data that I have on the quipu-\ike string records of North-American Indians indicate that there are two types. One is a long cord with knots and 1 The data on Peru knot records given here are drawn from a most interesting work, The Ancient Quipu or Peruvian Knot Record, by L. Leland Locke (American Museum of Natural History, 1923). Our photographs are from the frontispiece and from the diagram facing p. 16. See Figs. 16 and 17. 2 Erland Nordenskiold, Comparative Ethnographical Studies, No. 6, Part 1 (1925), p. 36. OLD NUMERAL SYMBOLS 39 bearing beads, etc., to indicate the days. It is simply a string record. This is known from the Yakima of eastern Washington and some In- terior Salish group of Nicola Valley, 1 B.C. FIG. 16. A quipu, from ancient Chancay in Peru, now kept in the American Museum of Natural History (Museum No. B8713) in New York City. 1 J. D. Leechman and M. R. Harrington, String Records of the Northwest, Indian Notes and Monographs (1921). 40 A HISTORY OF MATHEMATICAL NOTATIONS "The other type I have seen in use among the Havasupai and Walapai of Arizona. This is a cord bearing a number of knots to indi- cate the days until a ceremony, etc. This is sent with the messenger who carries the invitation. A knot is cut off or untied for each day that elapses; the last one indicating the night of the dance. This is also used by the Northern and South- ern Maidu and the Miwok of Cali- fornia. 1 There is a mythical ref- erence to these among the Zufii of New Mexico. 2 There is a note on its appearance in San Juan Pueblo in the same state in the seventeenth century, which would indicate that its use Was widely known among the Pueblo Indians. 'They directed him (the leader of the Pueblo rebellion of 1680) to make a rope of the palm leaf and tie in it a number of knots to rep- resent the number of days be- fore the rebellion was to take place; that he must send the rope to all the Pueblos in the Kingdom, when each should sig- nify its approval of, and union with, the conspiracy by untying one of the knots/ 3 The Huichol of Central Mexico also have knot- ted strings to keep count of days, untieing them as the days elapse. They also keep records of their lovers in the same way. 4 The Zufii also keep records of days worked in this fashion. 6 1 R. B. Dixon, "The Northern Maidu," Bulletin of the American Museum of Natural History, Vol. XVII (1905), p. 228, 271 ; P.-L. Faye, "Notes on the Southern Maidu," University of California Publications of American Archaeology and Ethnology, Vol. XX (1923), p. 44; Stephen Powers, "Tribes of California/' Contri- butions to North American Ethnology, Vol. Ill (1877), p. 352. 2 F. H. Gushing, "Zufti Breadstuff," Indian Notes and Monographs, Vol. VIII (1920), p. 77. 3 Quoted in J. G. Bourke, "Medicine-Men of the Apache," Ninth Annual Report, Bureau of American Ethnology (1892), p. 555. 4 K. Lumholtz, Unknown Mexico, Vol. II, p. 218-30. 6 Leechman and Harrington, op. cit. <** FIG. 17. Diagram of the two right- hand groups of strands in Fig. 16. OLD NUMERAL SYMBOLS 41 "Bourke 1 refers to medicine cords with olivella shells attached among the Tonto and Chiricahua Apache of Arizona and the Zufii. This may be a related form. "I think that there can be no question the instances of the second type are historically related. Whether the Yakima and Nicola Valley usage is connected with these is not established. " AZTECS 66. "For figures, one of the numerical signs was the dot (), which marked the units, and which was repeated either up to 20 or up to the figure 10, represented by a lozenge. The number 2Q was represented by a flag, which, repeated five times, gave the number 100, which was .:: O P Xi^Jto 10 &* 6 fVO 100 3t>0 $jjk *s || IP IP u I . "I " 1 1 1 I FIG. 18. Aztec numerals marked by drawing quarter of the barbs of a feather. Half the barbs was equivalent to 200, three-fourths to 300, the entire feather to 400. Four hundred multiplied by the figure 20 gave 8,000, which had a purse for its symbol." 2 The symbols were as shown in the first line of Figure 18. The symbols for 20, 400, and 8,000 disclose the number 20 as the base of Aztec numeration; in the juxtaposition of symbols the additive principle is employed. This is seen in the second line 3 of Figure 18, which represents 2X8,000+400+3X20+3X5+3 = 16,478 . 67. The number systems of the Indian tribes of North America, while disclosing no use of a symbol for zero nor of the principle of 1 Op. cit. y p. 550 ff. 2 Lucien Biart, The Aztecs (trans. 3. L. Garner; Chicago, 1905), p. 319. 8 Consult A. F. Pott, Die quindre und vigesimale Zdhlmethode bei Volkern aller Welttheile (Halle, 1847). 42 A HISTORY OF MATHEMATICAL NOTATIONS ***.** +*** " ' FIG. 19. From the Dresden Codex, of the Maya, displaying numbers. The second column on the left, from above down, displays the numbers 9, 9, 16, 0, 0, which stand for 9X 144,000+9X7,200+16 X360-fO+0 = 1,366,560. In the third column are the numerals 9, 9, 9, 16, 0, representing 1,364,360. The original appears in black and red colors. (Taken from Morley, An Introduction to the Study of the Maya Hieroglyphs, p. 266.) OLD NUMERAL SYMBOLS 43 local value, are of interest as exhibiting not only quinary, decimal, and vigesimal systems, but also ternary, quaternary, and octonary sys- tems. 1 MAYA 68. The Maya of Central America and Southern Mexico developed hieroglyphic writing, as found on inscriptions and codices, dating apparently from about the beginning of the Christian Era, which dis- closes the use of a remarkable number system and chronology. 2 The number system discloses the application of the principle of local value, and the use of a symbol for zero centuries before the Hindus began to use their symbol for zero. The Maya system was vigesimal, except in one step. That is, 20 units (kins, or "days") make 1 unit of the next higher order (uinals, or 20 days), 18 uinals make 1 unit of the third order (tun, or 360 days), 20 tuns make 1 unit of the fourth order (Katun, or 7,200 days), 20 Katuns make 1 unit of the fifth order (cycle, or 144,000 days), and finally 20 cycles make 1 great cycle of 2,880,000 days. In the Maya codices we find symbols for 1-19, expressed by bars and dots. Each bar stands for 5 units, each dot for 1 unit. For instance, ^- = . 1245 7 11 19 The zero is represented by a symbol that looks roughly like a half- closed eye. In writing 20 the principle of local value enters. It is expressed by a dot placed over the symbol for zero. The numbers are written vertically, the lowest order being assigned the lowest position (see Fig. 19). The largest number found in the codices is 12,489,781. CHINA AND JAPAN 69. According to tradition, the oldest Chinese representation of number was by the aid of knots in strings, such as are found later among the early inhabitants of Peru. There are extant two Chinese tablets 3 exhibiting knots representing numbers, odd numbers being designated by white knots (standing for the complete, as day, warmth, 1 W. C. Eells, "Number-Systems of North-American Indians, 7 ' American Mathematical Monthly, Vol. XX (1913), p. 263-72, 293-99; also Bibliotheca mathe- matica (3d series, 1913), Vol. XIII, p. 218-22. 2 Our information is drawn from S. G. Morley, An Introduction to the Study of the Maya Hieroglyphs (Washington, 1915). 3 Paul Perrty, Grammaire de la langue chinoise orale et ecrite (Paris, 1876), Vol. II, p. 5-7; Cantor, Vorlesungen iiber Geschichte der Mathematik, Vol. I (3d ed.), p. 674. 44 A HISTORY OF MATHEMATICAL NOTATIONS the sun) while even numbers are designated by black knots (standing for the incomplete, as night, cold, water, earth). The left-hand tablet shown in Figure 20 represents the numbers 1-10. The right-hand tablet pictures the magic square of nine cells in which the sum of each row, column, and diagonal is 15. 70. The Chinese are known to have used three other systems of writing numbers, the Old Chinese numerals, the mercantile numerals, and what have been designated as scientific numerals. The time of the introduction of each of these systems is uncertain. o 6 o o FIG. 20. Early Chinese knots in strings, representing numerals 71. The Old Chinese numerals were written vertically, from above down. Figure 21 shows the Old Chinese numerals and mercantile numerals, also the Japanese cursive numerals. 1 72. The Chinese scientific numerals are made up of vertical and horizontal rods according to the following plan : The numbers 1-9 are represented by the rods |, ||, |||, ||||, |||||, J, JL IJi Iffi; the numbers 10-90 are written thus _ = = = = ._!_=]= = == According to the Chinese author Sun-Tsu, units are represented, as just shown, by vertical rods, ten's by horizontal rods, hundred's again by vertical rods, and so on. For example, the number 6,728 was designated by 73. The Japanese make use of the Old Chinese numerals, but have two series of names for the numeral symbols, one indigenous, the other derived from the Chinese language, as seen in Figure 21. 1 See also Ed. Biot, Journal asiatique (December, 1839), p. 497-502; Cantor, Vorlesungen uber Geschichte der Maihematik, Vol. I, p. 673; Biernatzki, Crelle's Journal, Vol. LII (1856), p. 59-94. HINDU-ARABIC NUMERALS HINDU-ARABIC NUMERALS 45 74. Introduction. It is impossible to reproduce here all the forms of our numerals which have been collected from sources antedating 1500 or 1510 A.D. G. F. Hill, of the British Museum, has devoted a CHINOIS CH1FFKES VALEUBS. NOMS DE NOMBttE JAPOIfAIS cursifs. DU COMMERCE. EN > JAP01UIS PUR. Elf SINICO-JAPOBTAIS. 1 1 ftots. itsi. ^ ^ f( foutats. ni. H ^5. W 3 mils. san. E3 w^3 ^ l\ . yots. si. 3L rfS ^r 5 itsouts. g* -^ ^ _t. 6 mouts. rok. -t -t j. 7 nanats. silsi. A A 8 yatfi. fats. A ^l 3 9 kolconots. kou. -f* -f t 10 towd. zyou. lif "S ^ 100 moino. fakoufyak. =f* ^ f 1,000 teidzi. sen. M * ^ 10,000 yorodz. man. Fie. 21. Chinese and Japanese numerals. (Taken from A. P. Pihan, Expose des signes de numeration [Paris, 1860], p. 15.) whole book 1 of 125 pages to the early numerals in Europe alone. Yet even Hill feels constrained to remark: "What is now offered, in the shape of just 1,000 classified examples, is nothing more than a vinde- 1 The Development of Arabic Numerals in Europe (exhibited in 64 tables; Oxford, 1915). 46 A HISTORY OF MATHEMATICAL NOTATIONS miatio prima" Add to the Hill collection the numeral forms, or sup- posedly numeral forms, gathered from other than European sources, and the material would fill a volume very much larger than that of Hill. We are compelled, therefore, to confine ourselves to a few of the more important and interesting forms of our numerals. 1 75. One feels the more inclined to insert here only a few tables of numeral forms because the detailed and minute study of these forms has thus far been somewhat barren of positive results. With all the painstaking study which has been given to the history of our numerals we are at the present time obliged to admit that we have not even settled the time and place of their origin. At the beginning of the present century the Hindu origin of our numerals was supposed to have been established beyond reasonable doubt. But at the present time several earnest students of this perplexing question have ex- pressed grave doubts on this point. Three investigators G. II. Kaye in India, Carra de Vaux in France, and Nicol. Bubnov in Russia working independently of one another, have denied the Hindu origin. 2 However, their arguments are far from conclusive, and the hypothesis of the Hindu origin of our numerals seems to the present writer to explain the known facts more satisfactorily than any of the substitute hypotheses thus far advanced. 3 1 The reader who desires fuller information will consult Hill's book which is very rich in bibliographical references, or David Eugene Smith and Louis Charles Karpinski's The Hindu-Arabic Numerals (Boston and London, 1911). See also an article on numerals in English archives by H. Jenkinson in Antiquaries Journal, Vol. VI (1926), p. 263-75. The valuable original researches due to F. Woepcke should be consulted, particularly his great "Mdmoirc sur la propagation des chiffres indiens" published in the Journal asiatique (6th series; Paris, 1863), p. 27- 79, 234-90, 442-529. Reference should be made also to a few other publications of older date, such as G. Friedlein's Zahlzcichen und das elementare Rechnen der Griechen und Homer (Erlangen, 1869), which touches questions relating to our numerals. The reader will consult with profit the well-known histories of mathe- matics by H. Hankel and by Moritz Cantor. 2 G. R. Kaye, "Notes on Indian Mathematics," Journal and Proceedings of the Asiatic Society of Bengal (N.S., 1907), Vol. Ill, p. 475-508; "The Use of the Abacus in Ancient India," ibid., Vol. IV (1908), p. 293-97; "References to Indian Mathe- matics in Certain Mediaeval Works," ibid., Vol. VII (1911), p. 801-13; "A Brief Bibliography of Hindu Mathematics," ibid., p. 679-86; Scientia, Vol. XXIV (1918), p. 54; "Influence grecque dans le de"veloppement des mathc'matiques hindoues," ibid., Vol. XXV (1919), p. 1-14; Carra de Vaux, "Sur 1'origine des chiffres," ibid., Vol. XXI (1917), p. 273-82; Nicol. Bubnov, Arithmetische Selbst- stdndigkeit der europdischen Kultur (Berlin, 1914) (trans, from Russian cd.; Kiev, 1908). 3 F. Cajori, "The Controversy on the Origin of Our Numerals," Scientific Monthly, Vol. IX (1919), p. 458-64. See also B. Da'tta in Amer. Math. Monthly, Vol. XXXIII, p. 449; Proceed. Benares Math. Soc. t Vol. VII. HINDU-ARABIC NUMERALS 47 76. Early Hindu mathematicians, Aryabhata (b. 476 A.D.) and Brahmagupta (b. 598 A.D.), do not give the expected information about the Hindu-Arabic numerals. Aryabhata's work, called Aryabhatiya, is composed of three parts, in only the first of which use is made of a special notation of numbers. It is an alphabetical system 1 in which the twenty-five consonants represent 1-25, respectively; other letters stand for 30, 40, . . . . , 100, etc. 2 The other mathematical parts of Aryabhata consists of rules without examples. Another alphabetic system prevailed in Southern India, the numbers 1-19 being designated by consonants, etc. 3 In Brahmagupta's Pulverizer, as translated into English by H. T. Colebrooke, 4 numbers are written in our notation with a zero and the principle of local value. But the manuscript of Brahmagupta used by Colebrooke belongs to a late century. The earliest commentary on Brahmagupta belongs to the tenth century; Colebrooke's text is later. 5 Hence this manuscript cannot be accepted as evidence that Brahmagupta himself used the zero and the principle of local value. 77. Nor do inscriptions, coins, and other manuscripts throw light on the origin of our numerals. Of the old notations the most impor- tant is the Brahmi notation which did not observe place value and in which 1, 2, and 3 are represented by , , = . The forms of the Brahmi numbers do not resemble the forms in early place-value nota- tions 6 of the Hindu-Arabic numerals. Still earlier is the Kharoshthi script, 7 used about the beginning of the Christian Era in Northwest India and Central Asia. In it the first three numbers are I II III, then X = 4, IX = 5, IIX = 6, XX = 8, 1 = 10, 3 = 20, 33=40, 133 = 50, XI =100. The writing proceeds from right to left. 78. Principle of local value. Until recently the preponderance of authority favored the hypothesis that our numeral system, with its concept of local value and our symbol for zero, was wholly of Hindu origin. But it is now conclusively established that the principle of 1 M. Cantor, Vorlesungen liber Geschichte der Malhematik, Vol. I (3d ed.), p. 606. 2 G. R. Kaye, Indian Mathematics (Calcutta and Simla, 1915), p. 30, gives full explanation of Aryabhata's notation. 8 M. Cantor, Math. Beitrdge z. Kulturkben der Volkcr (1863), p. 68, 69. 4 Algebra with Arithmetic and Mensuration from the Sanscrit (London, 1817), p. 326 ff. 6 Ibid., p. v, xxxii. 6 See forms given by G. R. Kaye, op. cit., p. 29. 7 Ibid. 48 A HISTORY OF MATHEMATICAL NOTATIONS local value was used by the Babylonians much earlier than by the Hindus, and that the Maya of Central America used this principle and symbols for zero in a well-developed numeral system of their own and at a period antedating the Hindu use of the zero ( 68). 79. The earliest-known reference to Hindu numerals outside of India is the one due to Bishop Severus Sebokht of Nisibis, who, living in the convent of Kenneshre on the Euphrates, refers to them in a fragment of a manuscript (MS Syriac [Paris], No. 346) of the year 662 A.D. Whether the numerals referred to are the ancestors of the modern numerals, and whether his Hindu numerals embodied the principle of local value, cannot at present be determined. Apparently hurt by the arrogance of certain Greek scholars who disparaged the Syrians, Sebokht, in the course of his remarks on astronomy and mathematics, refers to the Hindus, " their valuable methods, of cal- culation ; andjjieir computing that surpasses description. Ijwish only to say that this computation is done by means of nine signs." 1 80. Some interest attaches to the earliest dates indicating the use of the perfected Hindu numerals. That some kind of numerals with a earlier than the ninth century is indicated by Brahmagupta (b. 598 A.D.), who gives rules for computing with a #ero. 2 G. Biihler 3 believes he has found definite mention of the decimal system and zero m the year 620 A.D. These statements do not neces- sarily imply the use of a decimal" system based on the principle of local value. G. R. Kaye 4 points out that the task of the antiquarian is complicated by the existence of forgeries. In the eleventh century in India "there occurred a specially great opportunity to regain con- fiscated endowments and to acquire fresh ones." Of seventeen cita- tions of inscriptions before the tenth century displaying the use of place value in writing numbers, all but two are eliminated as forgeries; these two are for the years 813 and 867 A.D.; Kaye is not sure of the reliability even of these. According to D.JE. Smith _and JLjg.^ Kar- pinski, 5 the earliest authentic document unmistakably containing the numerals mttMyh^^r^njMia belongs to the year 876 A.D. The earli- 1 See M. F. Nau, Journal asiatique (10th ser., 1910), Vol. XVI, p. 255; L. C. Karpinski, Science, Vol. XXXV (1912), p. 969-70; J. Ginsburg, Bulletin of the American Mathematical Society, Vol. XXIII (1917), p. 368. 2 Colebrooke, op. cit. r p. 339, 340. 3 "Indische Palaographie," Grundriss d. indogerman. Philologie u. Alieriuma- kunde, Band I, Heft 11 (Strassburg, 1896), p. 78. 4 Journal of the Asiatic Society of Bengal (N.S., 1907), Vol. Ill, p. 482-87. 5 The Hindu-Arabic Numerals (New York, 1911), p. 52. HINDU-ARABIC NUMERALS 49 est Arabic manuscripts containing the numerals are c>f_874 l and 888 A.D. They appear again in a work written at Shiraz in Persia 2 in 970 A.D. A church pillar 3 not far from the Jeremias Monastery in Egypt has I a 3 4 5 6 7 8 9 10 it la '3 I T I I I 1 i t * 7 I t I 1 CD r IT "6 T T t S er r z & r M 5 Ih rt u B fifi V * Q b 1? S H O *r b b K b t b L T V /t A V V V v V A X A yy, 8 3 a 8 8 sr 8 8 6 8 B & 8 8 8 I x S 2 / 6 CO 5 S> 976 x 1077 Ixi XI XI or XII | beg. XII? XII XII* C. 1200 C. 1200 XII XV XVI early FIG. 22. G. F. Hill's table of early European forms and Boethian apices. (From G. F. Hill, The Development of Arabic Numerals in Europe [Oxford, 1915], p. 28. Mr. Hill gives the MSS from which the various sets of numerals in this table are derived: [1] Codex Vigilanus; [2] St. Gall MS now in Zurich; [3] Vatican MS 3101, etc. The Roman figures in the last column indicate centuries.) 1 Karabacek, Wiener Zeitschriftfur die Kunde des Morgenlandcs, Vol. II (1897), p. 56. 2 L. C. Karpinski, Bibliotheca mathemalica (3d ser., 1910-11), p. 122. 3 Smith and Karpinski, op. cit., p. 138-43. ut Ant*. NommliofMui- 50 A HISTORY OF MATHEMATICAL NOTATIONS the date 349 A.H. ( = 961 A.D.). The oldest definitely dated European manuscript known to contain the Hindu-Arabic numerals is the Codex Vigilanus (see Fig. 22, No. 1), written in the Albelda Cloister in Spain in 976 A.D. The nine characters without the zero are given, as an addition, in a Spanish copy of the Origines by Isidorus of Seville, 992 A.D. A tenth-century manuscript with forms differing materially from those in the Codex Vigilanus was found in the St. Gall manu- script (see Fig. 22, No. 2), now in the University Library at Zurich. [The numerals are contained in a Vatican manuscript of 1077 (see Fig. 22, No. 3), on a Sicilian coin of 1138, in a Regensburg (Bavaria) ssi^r $^ uu M>f* n ApfOM Of Boothia* _. x ^> r A. x-> *B4 of tho MUdlo | "* ^k f\ f 9 Cl [3 ^/\ g 5 Sr.-r- 1 iT7*9- c fr t /'?8c J> Nam.rU, ofthoj p JXfjiortfoOT/Jn V A <\ I l>^}afiJVA3 t^^tf'H^fcr^o lyonthoJffmwr ^J 9 jf /" x^ A P Q C> gSKtr- l 2 3 f y * \ * 3 JTSJir; i * 3~3*4<i5 6 A~7 890 WMHor(T).148. ^ v/^l|wv/\ v j ^ From /> ^rl -_ .g ^ x *%. > . s P ptfln<r byi i. 54-5 6 78 ^10 TODlUU, 1M. ^J< / w /*^ FIG. 23. Table of important numeral forms. (The first, six lines in this table are copied from a table at the end of Cantor's Vorlesungen liber Geschichte der Mathematik, Vol. 1. The numerals in the Bamberg arithmetic are taken from Friedrich linger, Die Methodik der praktischen Arithmetik in historischer Eni- wickelung [Leipzig, J88S], p. 39.) chronicle of 1197. The earliest manuscript in French giving the numerals dates about 1275. In the British Museum one English manu- script is of about 1230-50; another is of 1246. The earliest undoubted Hindu-Arabic numerals on a gravestone are at Pforzheim in Baden of 1371 and one at Ulm of 1388. The earliest coins outside of Italy that are dated in the Arabic numerals are as follows: Swiss 1424, Austrian 1484, French 1485, German 1489, Scotch 1539, English 1551. 81. Forms of numerals. The Sanskrit letters of the second cen- tury A.D. head the list of symbols in the table shown in Figure 23. The implication is that the numerals have evolved from these letters. If such a connection could be really established, the Hindu origin of our numeral forms would be proved. However, a comparison of the forms appearing in that table will convince most observers that an origin HINDU-ARABIC NUMERALS 61 from Sanskrit letters cannot be successfully demonstrated in that way; the resemblance is no closer than it is to many other alphabets. The forms of the numerals varied considerably. The 5 was the most freakish. An upright 7 was rare in the earlier centuries. The symbol for zero first used by the Hindus was a dot. 1 The symbol for zero (0) of the twelfth anHTEIrteenth centuries i is ^sometiines crossed b^ a horizontal line, or a line slanting upward. 2 The Boethian apices, as found in some manuscripts, contain a triangle inscribed in the circular zero. In Athelard of Bath's translation of Al-Madjrltl's re- vision of Al-Khowarizmi's astronomical tables there are in different manuscripts three signs for zero, 3 namely ,_JhejQ ( = theta?) referred to above, then T ( = teca),* and 0. In oncToFlKe manuscripts 38 is written severaTtlmes XXXO, and 28 is written XXO, the being intended most likely as the abbreviation for oclo ("eight")- 82. The symbol T for zero is found also in a twelfth-century manuscript 5 of N. Ocreatus, addressed to his master Athelard. In that century it appears especially in astronomical tables as an ab- breviation for tcca, which, as already noted, was one of several names for zero; 6 it is found in those tables by itself, without connection with other numerals. The symbol occurs in the Alyorixmus vulyaris as- cribed to Sacrobosco. 7 C. A. Nallino found o for zero in a manuscript of Escurial, used in the preparation of an edition of Al-Battani. The [symbol for zero occurs also in printed mathematical books. The one author who in numerous writings habitually used 6 for zero was the French mathematician Michael Rollc (1652-1719). One finds it in his Traite d'algebre (1690) and in numerous articles in the publications of the French Academy and in the Journal des s^avans. 1 Smith and Karpinski, op. cit., p. 52, 53. 2 Hill, op. cit., p. 30-60. 3 II. Suter, Die astronomischen Tafeln des Muhammed ibn Musd Al-Khwdrizml in der Bearbeitang des Maslama ibn Ahmed Al-Madjrltl und der lateinischen Uebcr- setzung des Alhelhard von Bath (K^benhavn, 1914), p. xxiii. 4 See also M. Curtze, Petri Philomeni de Dacia in Algorismum vulgarem Johannis de Sacrobosco Commentarius (Hauniae, 1897), p. 2, 20. 6 "Prologus N. Ocreati in Helceph ad Adelardum Batcnscm Magistrum suum. Fragment sur la multiplication et la division public* pour la premiere fois par Charles Henry," Abhandlungen zur Geschichle der Mathematik, Vol. Ill (1880), p. 135-38. 6 M. Curtze, Urkunden zur Geschichte der Mathematik im Mittelalter und der Renaissance (Leipzig, 1902), p. 182. 7 M. Curtze, Abhandlungen zur Geschichte der Mathematik, Vol. VIII (Leipzig, 1898), p. 3-27. 52 A HISTORY OF MATHEMATICAL NOTATIONS Manuscripts of the fifteenth century, on arithmetic, kept in the Ashmolean Museum 1 at Oxford, represent the zero by a circle, crossed by a vertical stroke and resembling the Greek letter <. Such forms for zero arc reproduced by G. F. Hill 2 in many of his tables of numer- als. 83. In the fifty-six philosophical treatises of the brothers Ibwan as-safa (about 1000 A.D.) are shown Hindu-Arabic numerals and the corresponding Old Arabic numerals. The forms of the Hindu-Arabic numerals, as given in Figure 24, have maintained themselves in Syria to the present time. They ap- pear with almost identical form in an Arabic school primer, printed i : ? : * y 1 : f r I f> L: t: * u 1 A v i f r r t Fir,. 24. In the first line are the Old Arabic numerals for 10, 9, 8, 7, 6, 5, 4, 3, 2, 1. In the second line are the Arabic names of the numerals. In the third line are the Hindu-Arabic numerals as given by the brothers Ifrwan as-safa. (Repro- duced from J. Ruska, op. cit., p. 87.) at Beirut (Syria) in 1920. The only variation is in the 4, which in 1920 assumes more the form of a small Greek epsilon. Observe that is represented by a dot, and 5 by a small circle. The forms used in mod- ern Arabic schoolbooks cannot be recognized by one familiar only with the forms used in Europe. 84. In fifteenth-century Byzantine manuscripts, now kept in the Vienna Library, 3 the numerals used are the Greek letters, but the principle of local value is adopted. Zero is 7 or in some places ; aa means 11, py means 20, ayyy means 1,000. "This symbol 7 for zero means elsewhere 5," says Heiberg, "conversely, o stands for 5 (as now among the Turks) in Byzantine scholia to Euclid ..... In Constanti- nople the new method was for a time practiced with the retention of 1 Robert Stcclc, The Earliest Arithmetics in English (Oxford, 1922), p. 5. 2 Op. cit., Tables III, IV, V, VI, VIII, IX, XI, XV, XVII, XX, XXI, XXII. See also E. Wappler, Zur Geschichte der deutschen Algebra im XV. Jahrhundert (Zwickauer Gymnasialprogramm von 1887), p. 11-30. 3 J. L. Heiberg, "Byzantinische Analekten," Abhandlungen zur Geschichte der Mathematik, Vol. IX (Leipzig, 1899), p. 163, 166, 172. This manuscript in the Vienna Library is marked "Codex Phil. Gr. 65." HINDU-ARABIC NUMERALS 53 the old letter-numerals, mainly, no doubt, in daily intercourse." At the close of one of the Byzantine manuscripts there is a table of numerals containing an imitation of the Old Attic numerals. The table gives also the Hindu-Arabic numerals, but apparently without recog- nition of the principle of local value; in writing 80, the is placed over the 8. This procedure is probably due to the ignorance of the scribe. 85. A manuscript 1 of the twelfth century, in Latin, contains the symbol h for 3 which Curtze and Nagl 2 declare to have been found only in the twelfth century. According to Curtze, the foregoing strange symbol for 3 is simply the symbol for tertia used in the nota- tion for sexagesimal fractions which receive much attention in this manuscript. 86. Recently the variations in form of our numerals have been sum- marized as follows: "The form 3 of the numerals 1, 6, 8 and 9 has not varied much among the [medieval] Arabs nor among the Christians of the Occident; the numerals of the Arabs of the Occident for 2, 3 and 5 have forms offering some analogy to ours (the 3 and 5 are originally reversed, as well among the Christians as among the Arabs of the Occident); but the form of 4 and that of 7 have greatly modified themselves. The numerals 5, 6, 7, 8 of the Arabs of the Orient differ distinctly from those of the Arabs of the Occident (Gobar numerals). For five one still writes 5 and _J." The use of i for 1 occurs in the first printed arithmetic (Treviso, 1478), presumably because in this early . stage of printing there was no type for 1. Thus, 9,341 was printed 934;. 87. Many points of historical interest are contained in the fol- lowing quotations from the writings of Alexander von Humboldt. Although over a century old, they still are valuable. "In the Gobar 4 the group signs are dots, that is zeroes, for in India, Tibet and Persia the zeroes and dots are identical. The Gobar symbols, which since the year 1818 have commanded my whole at- tention, were discovered by my friend and teacher, Mr. Silvestre de Sacy, in a manuscript from the Library of the old Abbey St. Germain du Pres. This great orientalist says: 'Le Gobar a un grand rapport 1 Algorithmus-MSS Clm 13021, fols. 27-29, of the Munich Staatsbibliothek. Printed and explained by Maximilian Curtze, Abhandlungen zur Geschichte der Mathematik, Vol. VIII (Leipzig, 1898), p. 3-27. 2 Zeilschrift fur Mathematik und Physik (Hist. Litt. Abth.), Vol. XXXIV (Leipzig, 1889), p. 134. 3 Encyc. des Stien. math., Tome I, Vol. I (1904), p. 20, n. 105, 106. 4 Alexander von Humboldt, Crelle's Journal, Vol. IV (1829), p. 223, 224. 54 A HISTORY OF MATHEMATICAL NOTATIONS avec le chifYre indien, mais 11 n'a pas de z6ro (S. Gramm. arabe, p. 76, and the note added to PL 8).' I am of the opinion that the zero- symbol is present, but, as in the Scholia of Neophytos on the units, it stands over the units, not by their side. Indeed it is these very zero- symbols or dots, which give these characters the singular name Gobar or dust-writing. At first sight one is uncertain whether one should recognize therein a transition between numerals and letters of the alphabet. One distinguishes with difficulty the Indian 3, 4, 5 and 9. Dal and ha are perhaps ill-formed Indian numerals 6 and 2. The nota- tion by dots is as follows: 3 ' for 30 , 4" for 400, 6 * for 6,000 . These dots remind one of an old-Greek but rare notation (Ducarige, Palacogr., p. xii), which begins with the myriad: a" for 10,000, fl :: for 200 millions. In this system of geometric progressions a single dot, which however is not written down, stands for 100. In Diophantus and Pappus a dot is placed between letter-numerals, instead of the initial Mv (myriad). A dot multiplies what lies to its left by 10,000. .... A real zero symbol, standing for the absence of some unit, is ap- plied by Ptolemy in the descending sexagesimal scale for missing de- grees, minutes or seconds. Delambre claims to have found our sym- bol for zero also in manuscripts of Theon, in the Commentary to the Syntaxis of Ptolemy. 1 It is therefore much older in the Occident than the invasion of the Arabs and the work of Planudes on arithmoi indikoi." L. C. Karpinski 2 has called attention to a passage in the Arabic biographical work, the Fihrist (987 A.D.), which describes a Hindu notation using dots placed below the numerals; one dot indi- cates tens, two dots hundreds, and three dots thousands. 88. There are indications that the magic power of the principle of local value was not recognized in India from the beginning, and that our perfected Hindu-Arabic notation resulted from gradual evolution. Says Hurnboldt: "In favor of the successive perfecting of the designa- tion of numbers in India testify the Tamul numerals which, by means 1 J. B. J. Delambre, Histoire de Vastron. ancienne, Vol. I, p. 547; Vol. II, p. 10. The alleged passage in the manuscripts of Theon is not found in his printed works. Delambre is inclined to ascribe the Greek sign for zero either as an abbreviation of ouden or as due to the special relation of the numeral omicron to the sexagesimal fractions (op. cit., Vol. II, p. 14, and Journal des sgavans [1817], p. 539). *Bibliolheca malficmaiica, Vol. XI (1910-11), p. 121-24. HINDU-ARABIC NUMERALS 55 of the nine signs for the units and by signs of the groups 10, 100, or 1,000, express all values through the aid of multipliers placed on the r v +3 K^ : i5>ns- x i, -X.jjL.^|.^| ^^^xiiffi 'i > ^- mi% a o fl o a H left. This view is supported also by the singular arithmoi indikoi in the scholium of the monk Neophytos y which is found in the Parisian A HISTORY OF MATHEMATICAL NOTATIONS Qit tJifler jiffir werben 0<ontt<$ mit fren &* ractcrn vie btrrwfc rdgulfo gift ribcn/|>abe flUlcfr* volnirvil fonbcrc rcrn>anMun00r0cit ixngcmci* fwnjiffcrn/aufttnomen *>A f&nfft rn& ffoenfc. 4u$foU&u fbnberlufcnurcftn/ wenn bey cfncr 5i(fr6rtypon<tOebn/fobU5a(ftlbi0pag gcrrt fo Ml ?mcr/n& f tin turt^eil minbcr no$ mcbn 0ie balbcn 2?mcr tr<rt>en allein mit cincr Imi Oder f!ri4>lm 9nrerr<beit>cn.9efi ale off t cin flricfclin btircfr 4<n jiffer gcbi/bcnlmpt re tin balben 2^mer/pnb ba <jef<fei^t tUtin (>cy 5<n iymcrn pnnb ntcfai t>ein pur/ *1 li<b9<ertcl!mebto^er minber ober bit 0e4 funben jrmcrbetr/ ba wirbr bur^ die jtxxy vo!0cnben jty< n/ onb '!' fer b~tbtut\u vw .i^tfWeunbbalbfcymer. J O-f Jcbcntbalbtr ^ Btbeotbam'cr; -- \ ' let minder. JL* ! Nf * w S FIG. 26. From Christoff Ru- dolff's Kunstliche Rechnung mit der Ziffer (Augsburg, 1574[?j). Library (Cod. Reg., fol. 15), for an account of which I am indebted to Prof. Brandis. The nine digits of Neophytos wholly resemble the Per- sian, except the 4. The digits 1, 2, 3 and 9 are found even in Egyptian number inscriptions (Koscgarten, de Hierogl. AegypL, p. 54). The nine units are enhanced tenfold, 100 fold, 1,000 fold by writing above them one, two or three zeros, as in: o o o o oo 2-20, 24 = 24, 5 = 500, 6 = 6,000. If we imagine dots in place of the zero symbols, then we have the arabic Gobar numerals." 1 Humboldt copies the scholium of Neophytos. J. L. Heiberg also has called atten- tion to the scholium of Neophytos and to the numbering of scholia to Euclid in a Greek manuscript of the twelfth century (Codex Vindo- bonensis, Gr. 103), in which numer- als resembling the Gobar numerals occur. 2 The numerals of the monk Neophytos (Fig. 25), of which Humboldt speaks, have received the special attention of P. Tannery. 3 89. Freak forms. We reproduce herewith from the Augsburg edition of Christoff Rudolff's Kunstliche Rechnung a set of our numerals, and of symbols to represent such fractions 1 Op. tit., p. 227. 2 See J. L. Heiberg's edition of Euclid (Leipzig, 1888), Vol. V; P. Tannery, Revue archeol. (3d scr., 1885), Vol. V, p. 99, also (3d scr., 1886), Vol. VII, p. 355; Encyd des scien. math., Tome I, Vol. I (1904), p. 20, n. 102. 3 Memoir es scientifiques, Vol. IV (Tou- louse and Paris, 1920), p. 22. HINDU-ARABIC NUMERALS 57 and mixed numbers as were used in Vienna in the measurement of wine. We have not seen the first edition (1526) of Rudolff's book, but Alfred Nagl 1 reproduces part of these numerals from the first edition. "In the Viennese wine-cellars," says Hill, "the casks were marked according to their contents with figures of the forms given." 2 The symbols for fractions are very curious. 90. Negative numerals. J. Colson 3 in 1726 claimed that, by the use of negative numerals, operations may be performed with "more ease and expedition." If 8605729398715 is to be multiplied by 389175836438, reduce these to small numbers 1414331401315 and 4l 1224244442. Then write the multiplier on a slip of paper and place it in an inverted position, so that its first figure is just over the left-hand figure of the multiplicand. Multiply 4X1=4 and write down 4. Move the multiplier a place to the right and collect the two products, 4X1 + 1X1 = 5; write down 5. Move the multiplier another place to the right, then 4X4+1X1 + 1X1 = 16; write the 1 in the second line. Similarly, the next product is 11, and so on. Similar processes and notations were proposed by A. Cauchy, 4 E. Selling, 5 and W. B. Ford, 6 while J. P. Ballantino 7 suggests 1 inverted, thus i, as a sign for negative 1, so that 1X7 = 13 and the logarithm 9 . 69897 - 10 may be written 19 . 69897 or I . 69897. Negative logarithmic charac- teristics are often marked with a negative sign placed over the numeral (Vol. II, 476). 91. Grouping digits in numeration. In the writing of numbers con- taining many digits it is desirable to have some symbol separating the numbers into groups of, say, three digits. Dots, vertical bars, commas, arcs, and colons occur most frequently as signs of separation. In a manuscript, Liber algorizmi, 8 of about 1200 A.D., there appear 1 Monatsblalt der numismatischen Gesellschaft in Wien, Vol. VII (December, 1906), p. 132. 2 G. F. Hill, op. cU., p. 53. 3 Philosophical Transactions, Vol. XXXIV (1726), p. 161-74; Abridged Trans- actions, Vol. VI (1734), p. 2-4. See also G. Peano, Formulaire mathematique, Vol. IV (1903), p. 49. 4 Comptcs rendus, Vol. XI (1840), p. 796; (Euvres (1st ser.), Vol. V, p. 434-55. 6 Eine mue Rechenrnaschine (Berlin, 1887), p. 16; see also Encyklopddie d. Math. Wiss., Vol. I, Part 1 (Leipzig, 1898-1904), p. 944. 6 American Mathematical Monthly, Vol. XXXII (1925), p. 302. 7 Op. til., p. 302. 8 M. Cantor, Zeitschrift fur Mathematik, Vol. X (1865), p. 3; G. Enestrom, Bibliothcca mathematica (3d ser., 1912-13), Vol. XIII, p. 265. 58 A HISTORY OF MATHEMATICAL NOTATIONS dots to mark periods of three. Leonardo of Pisa, in his Liber Abbaci (1202), directs that the hundreds, hundred thousands, hundred mil- lions, etc., be marked with an accent above; that the thousands, millions, thousands of millions, etc., be marked with an accent below. In the 1228 edition, 1 Leonardo writes 678 935 784 105 296. Johannes de Sacrobosco (d. 1256), in his Tractatus de arte numerandi, suggests that every third digit be marked with a dot. 2 His commentator, Petrus de Dacia, in the first half of the fourteenth century, does the same. 3 Directions of the same sort are given by Paolo Dagomari 4 of Florence, in his Regoluzze di Maestro Paolo doll Abbaco and Paolo of Pisa, 5 both writers of the fourteenth century. Luca Pacioli, in his Summa (1494), folio 196, writes 8 659 421 635 894 676; Georg Peur- bach (1505), 6 "3790528614. Adam Riese 7 writes 86789325178. M. Stifel (1544) 8 writes 2329089562800. Gemma Frisius 9 in 1540 wrote 24 456 345 678. Adam Riese (1535) 10 writes 86 -7 -89 -3 -25 -178. The Dutch writer, Martinus Carolus Creszfeldt, 11 in 1557 gives in his Arithmetica the following marking of a number: "Exempei. || 5 8 7 4 9 3 6 2 5 3 4 || ." w i w i w i w 1 El liber abbaci di Leonardo Pisano .... da B. Boncompagni (Roma, 1857), p. 4. 2 J. O. Ilalliwcll, Rara malhematica (London, 1839), p. 5; M. Cantor, Vor- lesungen, Vol. II (2d cd., 1913), p. 89. 3 Petri Philomeni de Dacia in Algorismum vulgar em lohannis de Sacrobosco commentarius (ed. M. Curtze; Kopenhagen, 1897), p. 3, 29; J. Tropfke, Geschichte der Nlemcntarmathematik (2d cd., 1921), Vol. I, p. 8. 4 Libri, Histoire des sciences mathematiques en Italic, Vol. Ill, p. 296-301 (Rule 1). 6 Ibid., Vol. II, p. 206, n. 5, and p. 526; Vol. Ill, p. 295; see also Cantor, op. cit., Vol. II (2d ed., 1913), p. 164. 6 Opus alyorithmi (Herbipoli, 1505). See Wildermuth, "Rechnen," Encyklo- paedie des gesammten Erziehungs- und Unterrichtsivesens (Dr. K. A. Schmid, 1885). 7 Hechnung auff der Linien vnnd Federn (1544); Wildermuth, "Reehnen," Encijklopaedie (Schmid, 1885), p. 739. 8 Wildermuth, op. cit., p. 739. 9 Arithmetical practicae methodus facilis (1540) ; F. linger, Die Methodik der praktischen Arithmetik in hislorischer Entwickelung (Leipzig, 1888), p. 25, 71. 10 Rechnung auff d. Linien u. Federn (1535). Taken from H. Hankel, op. cit. (Leipzig, 1874), p. 15. 11 Arithmetica (1557). Taken from Bierens de Haan, Bouwstoffen voor de Ge- schiedenis der Wis-en Natuurkundige Wetenschappen t Vol. II (1887), p. 3. HINDU-ARABIC NUMERALS 59 Thomas Blundeville (1636) 1 writes 5|936|649. Tonstall 2 writes . ... 43210 3210987654321. Clavius 3 writes 42329089562800. Chr. Rudolff 4 writes 23405639567. Johann Caramuel 6 separates the digits, as in "34:252,- Integri. Partes. 341;154,329"; W. Oughtred, 6 9!876i543|210l2i345678i9; K. Schott 7 , 7697432329089562436; N. Barreme, 8 254.567.804.652; W. J. G. Ill II I Karsten, 9 872 094,826 152,870 364,008; I. A. de Segner, 10 5|329 // |870| 325 / |743|297, 174; Thomas Dilworth/ 1 789 789 789; Nicolas Pike, 12 3 2 1 356;809,379;120,406;129,763; Charles Hutton, 13 281,427,307; E. Bczout, 14 23, 456, 789, 234, 565, 456. In M. Lcmos' Portuguese encyclopedia 15 the population of New 1 Mr. Klundcvil, His Exercises contayning eight Treatises (7th cd., Ro. Hartwell; London, 1636), p. 106. 2 De Artc Svppvtandi, libri qvatvor Cvtheberti Tonstalli (Argentorati), Colophon 1544, p. 5. 3 Christophori Clavii epitome arithmeticae practicac (Romae, 1583), p. 7. 4 Kunstliche Rechnung mil dcr Ziffer (Augsburg, 1574[?J), Aiij B. 6 Joannis Caramvclis mathesis biceps, veins et nova (Cornpaniae [southeast of Naples], 1670), p. 7. The passage is as follows: "Punctum finale ( . ) est, quod poni- tur post unitatem: ut cum scribirnus 23. viginti tria. Comma (,) post millemirium scribitur . . . . ut cum scribimus, 23,424 Millcriarium & centenario -dis- tinguere alios populos docent Hispani, qui utuntur hoc charactere \f , . . . . Hypo- colon (;) millioncm a millcnario separat, ut cum scribimus 2;041,311. Duo puncta ponuntur post billioncm, sen millioncm millionum, videlicet 34:252,341;154,329." Caramuel was born in Madrid. For biographical sketch see Rcvista matemdtica Hispano-American, Vol. I (1919), p. 121, 178, 203. 6 Clavis mathematicae (London, 1652), p. 1 (1st ed., 1631). 7 Cursus mathematicus (Herbipoli, 1661), p. 23. 8 Arithm6tique (new ed.; Paris, 1732), p. 6. 9 Mathesis theoretica elementaris atqve svblimior (Rostochii, 1760), p. 195. 10 Elementa arithmelicae gcomelriae et calcvli geometrici (2d ed.; Halle, 1767), p. 13. 11 Schoolmaster's Assistant (22d ed.; London, 1784), p. 3. 12 New and Complete System of Arithmetic (Newburyport, 1788), p. 18. 18 " Numeration," Mathematical and Philosophical Dictionary (London, 1795). 14 Cours de malMmatiques (Paris, 1797), Vol. I, p. 6. 16 ' 'Portugal," Encyclopedia Portugueza ILlustrada . . . . de Maximiano Lemos (Porto). CO A HISTORY OF MATHEMATICAL NOTATIONS York City is given as "3 .437:202"; in a recent Spanish encyclopedia, 1 the population of America is put down as "150- 979,995." In the process of extracting square root, two early commentators 2 on Bhaskara's Lilavati, namely Rama-Crishna Deva and Gangad'hara (ca. 1420 A.D.), divide numbers into periods of two digits in this man- ner, 8 8 2 9. In finding cube roots Rama-Crishna Deva writes i i i 1953125. 92. The Spanish "calderon" In Old Spanish and Portuguese numeral notations there are some strange and curious symbols. In a contract written in Mexico City in 1649 the symbols "7U291e" and "VIIUCCXCIps" each represent 7,291 pesos. The U, which here re- sembles an that is open at the top, stands for "thousands." 3 I. B. Richman has seen Spanish manuscripts ranging from 1587 to about 1700, and Mexican manuscripts from 1768 to 1855, all containing symbols for "thousands" resembling U or D, often crossed by one or two horizontal or vertical bars. The writer has observed that after 1600 this U is used freely both with Hindu- Arabic and with Roman numerals; before 1600 the U occurs more commonly with Roman numerals. Karpinski has pointed out that it is used with the Hindu- Arabic numerals as early as 1519, in the accounts of the Magellan voyages. As the Roman notation does not involve the principle of local value, U played in it a somewhat larger role than merely to afford greater facility in the reading of numbers. Thus VIUCXV equals 6X1,000+115. This use is shown in manuscripts from Peru of 1549 and 1543, 4 in manuscripts from Spain of 1480 5 and 1429. 6 We have seen the corresponding type symbol for 1,000 in Juan Perez de Moya, 7 in accounts of the coming in the Real Casa de Moneda de 1 "America," Encyclopedia illmtrada segui Diccionario universal (Barcelona). 2 Colebrooke, op. cit., p. 9, 12, xxv, xxvii. 3 F. Cajori, "On the Spanish Symbol U for 'thousands/ " Bibliotheca mathe- matica, Vol. XII (1912), p. 133. 4 Carlos de Indias publicalas por primer a vez d Ministerio de Fomento (Madrid, 1877), p. 502, 543, facsimiles X and Y. 5 Jose Gorizalo dc las Casas, Ancdes de la Palcoc/rafia Espanola (Madrid, 1857), Plates 87, 92, 109, 110, 113, 137. Liciniano Saez, Demoslracidn Histdricadel verdadero valor de todas las monedaa que corrian en Caslilla duranle el Reynado del Senor Don Enrique III (Madrid, 1796), p. 447. See also Colomera y Rodriguez, Venancio, Paleoyrajia caslellana (1862). 7 Arilmelica practica (14th ed.; Madrid, 1784), p. 13 (1st ed., 1562). HINDU-ARABIC NUMERALS 61 Mexico (1787), in eighteenth-century books printed in Madrid, 1 in the Gazetas de Mexico of 1784 (p. 1), and in modern reprints of seventeenth-century documents. 2 In these publications the printed symbol resembles the Greek sampi 5 for 900, but it has no known connection with it. In books printed in Madrid 3 in 1760, 1655, and 1646, the symbol is a closer imitation of the written U, and is curiously made up of the two /small printed letters, I, f, each turned halfway around. The two inverted letters touch each other below, thus \f . Printed symbols representing a distorted U have been found also in some Spanish arithmetics of the sixteenth century, particularly in that of Gaspard de Texeda 4 who writes the number 103,075,102,300 in the Castellanean form c.iijU.75qs c.ijU300 and also in the algoristic form 103U075qs 102U300". The Spaniards call this symbol and also the sampi-like symbol a calderon. 5 A non-Spanish author who ex- plains the calderdn is Johann Caramuel, 6 in 1670. 93. The present writer has been able to follow the trail of this curious symbol U from Spain to Northwestern Italy. In Adriano Cappelli's Lexicon is found the following: "In the liguric documents of the second half of the fifteenth century we found in frequent use, to indicate the multiplication by 1,000, in place of M, an O crossed by a horizontal line." 7 This closely resembles some forms of our Spanish symbol U. Cappelli gives two facsimile reproductions 8 in 1 Liciniano Saez, op. cit. 2 Manuel Danvila, Boletin de la Real Academia de la Hisloria (Madrid, 1888), Vol. XII, p. 53. 3 Cuentas para lodas, compendia arilhmetico, e Histdrico . . . . su autor D. Manuel Recio, Oficial de la contadurfa general de postos del Rcyno (Madrid, 1760) ; Teatro Eclesidstico de la primitiva Iglesia de las Indias Occidentals .... el M. Gil Gonzalez Davila, su Coronista Mayor de las Indias, y de los Reynos de las dos Castillas (Madrid, 1655), Vol. II; Memorial, y Noticias Sacras, y reales del Imperio de las Indias Occidentales .... Escriuiale por el afio de 1646, Juan Diez de la Calle, Oficial Segundo de la Misma Secretaria. *Suma de Arithmetica pratica (Valladolid, 1546), fol. iiijr.; taken from D. E. Smith, History of Mathematics, Vol. II (1925), p. 88. The qs means quentos (cuen- tos, "millions")- 5 In Joseph Aladern, Diccionari popular de la Llengua Catalana (Barcelona, 1905), we read under "Caldero": "Among ancient copyists a sign (\/") denoted a thousand." 6 Joannis Caramvelis Mathesis biceps veins et nova (Companiae, 1670), p. 7. 7 Lexicon Abbreviaturarum (Leipzig, 1901), p. 1. 8 Ibid., p. 436, col. 1, Nos. 5 and 6. 62 A HISTORY OP MATHEMATICAL NOTATIONS which the sign in question is small and is placed in the position of an exponent to the letters XL, to represent the number 40,000. This corresponds to the use of a small c which has been found written to the right of and above the letters XI, to signify 1,100. It follows, there- fore, that the modified U was in use during the fifteenth century in Italy, as well as in Spain, though it is not known which country had the priority. What is the origin of this calderon? Our studies along this line make it almost certain that it is a modification of one of the Roman F ' A /* * !*' FIG. 27. From a contract (Mexi o City, 1649). The right part shows the sum of 7,291 pesos, 4 tomines, 6 granos, ex >ressed in Roman numerals and the calderdn. The left part, from the same contract, shows the same sum in Hindu-Arabic nu- merals and the calderdn. symbols for 1,000. Besides M, the Romans used for 1,000 the symbols CIO, T, oo, and *f . These symbols are found also in Spanish manu- scripts. It is easy to see how in the hands of successive generations of amanuenses, some of these might assume the forms of the calderdn. If the lower parts of the parentheses in the forms CIO or CIIO are united, we have a close imitation of the U, crossed by one or by two bars. HINDU-ARABIC NUMERALS 63 94. The Portuguese "cifrao." Allied to the distorted Spanish U is the Portuguese symbol for 1,000, called the cifrao. 1 It looks somewhat like our modern dollar mark, $. But its function in writing numbers was identical with that of the calderon. Moreover, we have seen forms of this Spanish "thousand" which need only to be turned through a right angle to appear like the Portuguese symbol for 1,000. Changes of that sort are not unknown. For instance, the Arabic numeral 5 appears upside down in some Spanish books and manuscripts as late as the eighteenth and nineteenth centuries. a FIG. 28. Real estate sale in Mexico City, 1718. The sum written here is 4,255 pesos. 95. Relative size of numerals in tables. Andr6 says on this point: "In certain numerical tables, as those of Schron, all numerals are of the same height. In certain other tables, as those of Lalande, of Cal- let, of Houel, of Dupuis, they have unequal heights: the 7 and 9 are prolonged downward; 3, 4, 5, 6 and 8 extend upward; while 1 and 2 do not reach above nor below the central body of the writing The unequal numerals, by their very inequality, render the long train of numerals easier to read; numerals of uniform height are less legible." 2 1 See the word cifrao in Antonio de Moraes Silva, Dice, de Lingua Portuguesa (1877); in Vieira, Grande Dice. Portuguez (1873); in Dice. Comtemp. da Lingua Portuguesa (1881). 2 D. Andre", Des notations math&matiques (Paris, 1909), p. 9. 64 A HISTORY OF MATHEMATICAL NOTATIONS 96. Fanciful hypotheses on the origin of the numeral forms. A p lem as fascinating as the puzzle of the origin of language relate the evolution of the forms of our numerals. Proceeding on the t assumption that each of our numerals contains within itself, { skeleton so to speak, as many dots, strokes, or angles as it repres units, imaginative writers of different countries and ages have vanced hypotheses as to their origin. Nor did these writers feel i they were indulging simply in pleasing pastime or merely contribu to mathematical recreations. With perhaps only one exception, 1 were as convinced of the correctness of their explanations as are ch squarers of the soundness of their quadratures. The oldest theory relating to the forms of the numerals is du the Arabic astrologer Aben Ragel 1 of the tenth or eleventh cent He held that a circle and two of its diameters contained the rcqu forms as it were in a nutshell. A diameter represents 1; a diam and the two terminal arcs on opposite sides furnished the 2. A glanc Part I of Figure 29 reveals how each of the ten forms may be evol from the fundamental figure. On the European Continent, a hypothesis of the origin from do the earliest. In the seventeenth century an Italian Jesuit wn Mario Bettini, 2 advanced such an explanation which was eag accepted in 1651 by Georg Philipp Harsdorffer 3 in Germany, ' said: "Some believe that the numerals arose from points or dots, in Part II. The same idea was advanced much later by Geo Dumesnil 4 in the manner shown in the first line of Part III. In cur writing the points supposedly came to be written as dashes, yielc forms resembling those of the second line of Part III. The two hori: tal dashes for 2 became connected by a slanting line yielding the n ern form. In the same way the three horizontal dashes for 3 were joi by two slanting lines. The 4, as first drawn, resembled the 0; but < fusion was avoided by moving the upper horizontal stroke inl 1 J. F. Weidler, De characteribus numerorum vulgaribus dissertatio mathcma critica (Wittembcrgae, 1737), p. 13; quoted from M. Cantor, Kulturleben der V\ (Halle, 1863), p. 60, 373. 2 Apiaria unwersae philosophiae, mathematicae, Vol. II (1642), Apiarium p. 5. See Smith and Karpinski, op. cit., p. 36. 3 Delitae mathematicae et physicae (Niirnberg, 1651). Reference from Sterner, Geschichte der Rechenkunst (Miinchen and Leipzig [1891]), p. 138, 52 4 "Note sur la forme des chifTres usuels," Revue archSologique (3d ser.; P 1890), Vol. XVI, p. 342-48. See also a critical article, "Pretendues notal Pythagoriennes sur Forigine de nos chiffres," by Paul Tannery, in his Mem scientifiques, Vol. V (1922), p. 8. HINDU-ARABIC NUMERALS 65 vertical position and placing it below on the right. To avoid con- founding the 5 and 6, the lower left-hand stroke of the first 5 was fc7S? 3 D 5 E B 2 2 J 4 A 7 X 'i '- H 5 G n o % s S a 5 B I 2 3 o 8 O S 88'S o s 2 O a FIG. 29. Fanciful hypotheses < n a S " 5 a o changed from a vertical to a horizontal position and placed at the top of the numeral. That all these changes were accepted as historical, 66 A HISTORY OF MATHEMATICAL NOTATIONS without an atom of manuscript evidence to support the different steps in the supposed evolution, is an indication that Baconian inductive methods of research had not gripped the mind of Dumesnil. The origin from dots appealed to him the more strongly because points played a role in Pythagorean philosophy and he assumed that our numeral system originated with the Pythagoreans. Carlos le-Maur, 1 of Madrid, in 1778 suggested that lines joining the centers of circles (or pebbles), placed as shown in the first line of Part IV, constituted the fundamental numeral forms. The explana- tion is especially weak in accounting for the forms of the first three numerals. A French writer, P. Voizot, 2 entertained the theory that originally a numeral contained as many angles as it represents units, as seen in Part V. He did not claim credit for this explanation, but ascribed it to a writer in the Gcnova Catholico Militarite. But Voizot did originate a theory of his own, based on the number of strokes, as shown in Part VI. Edouard Lucas 3 entertains readers with a legend that Solomon's ring contained a square and its diagonals, as shown in Part VII, from which the numeral figures were obtained. Lucas may have taken this explanation from Jacob Leupold 4 who in 1727 gave it as widely current in his day. The historian Moritz Cantor 5 tells of an attempt by Anton Miiller 6 to explain the shapes of the digits by the number of strokes necessary to construct the forms as seen in Part VIII. An eighteenth-century writer, Georg Wachter, 7 placed the strokes differently, somewhat as in Part IX. Cantor tells also of another writer, Piccard, 8 who at one time had entertained the idea that the shapes were originally deter- 1 Elcmentos de Matematica pura (Madrid, 1778), Vol. I, chap. i. J "Lcs chiffres arabes et leur origine," La nature (2d semestre, 1899), Vol. XXVII, p. 222. 3 L' Arithmelique amusanle (Paris, 1895), p. 4. Also M. Cantor, Kulturlcben der Volker (Halle, 1863), p. 60, 374, n. 116; P. Treutlcin, Geschichte unsercr Zahl- zeicJien (Karlsruhe, 1875), p. 16. 4 Theatrvm Arithmetico-Geometricvm (Leipzig, 1727), p. 2 and Table III. 6 Kullurleben der Volker, p. 59, 60. 6 Arilhmetik und Algebra (Heidelberg, 1833). See also a reference to this in P. Treutlein, op. tit. (1875), p. 15. 7 Naturae et Sctipturae Concordia (Lipsiae et Hafniae, 1752), chap. iv. 8 M6moire sur la forme et de la provenance des chiffres, Sociele Vaudoise des sciences nalurelles (stances du 20 Avril et du 4 Mai, 1859), p. 176, 184. M. Cantor reproduces the forms due to Piccard; see Cantor, Kidturleben, etc., Fig. 44. HINDU-ARABIC NUMERALS 67 mined by the number of strokes, straight or curved, necessary to express the units to be denoted. The detailed execution of this idea, as shown in Part IX, is somewhat different from that of Mliller and some others. But after critical examination of his hypothesis, Pic- card candidly arrives at the conclusion that the resemblances he pointed out are only accidental, especially in the case of 5, 7, and 9, and that his hypothesis is not valid. This same Piccard offered a special explanation of the forms of the numerals as found in the geometry of Boethius and known as the "Apices of Boethius." He tried to connect these forms with letters in the Phoenician and Greek alphabets (see Part X). Another writer whose explanation is not known to us was J. B. Reveillaud. 1 The historian W. W. R. Ball 2 in 1888 repeated with apparent ap- proval the suggestion that the nine numerals were originally formed by drawing as many strokes as there are units represented by the respective numerals, with dotted lines added to indicate how the writ- ing became cursive, as in Part XL Later Ball abandoned this ex- planation. A slightly different attempt to build up numerals on the consideration of the number of strokes is cited by W. Lietzmann. 3 A still different combination of dashes, as seen in Part XII, was made by the German, David Arnold Crusius, in 1746. 4 Finally, C. P. Sherman 5 explains the origin by numbers of short straight lines, as shown in Part XI11. "As time went on," he says, "writers tended more and more to substitute the easy curve for the difficult straight line and not to lift the pen from the paper between detached lines, but to join the two which we will call cursive writing." These hypotheses of the origin of the forms of our numerals have been barren of results. The value of any scientific hypothesis lies in co-ordinating known facts and in suggesting new inquiries likely to advance our knowledge of the subject under investigation. The hy- potheses here described have done neither. They do not explain the very great variety of forms which our numerals took at different times 1 Essai sur lea chiffrcs arabcs (Paris, 1883). Reference from Smith and Kar- pinski, op. cit., p. 36. 2 A Short Account of the History of Mathematics (London, 1888), p. 147. 3 Lusliges und Merkwurdiges von Zahlen und Formcn (Brcslau, 1922), p. 73, 74. lie found the derivation in Raether, Theorie und Praxis dcs Rcchcnunterrichts (1. Teil, 0. Aufl.; Brcslau, 1920), p. 1, who refers to H. von Jacobs, Das Volk der Sicbener-Zdhler (Berlin, 1896). 4 Anweisung zur Rechen-Kunsl (Halle, 1746), p. 3. 6 Mathematics Teacher, Vol. XVI (1923), p. 398-401. 68 A HISTORY OF MATHEMATICAL NOTATIONS and in different countries. They simply endeavor to explain the nu- merals as they are printed in our modern European books. Nor have they suggested any fruitful new inquiry. They serve merely as en- tertaining illustrations of the operation of a pseudo-scientific imagina- tion, uncontrolled by all the known facts. 97. A sporadic artificial system. A most singular system of numeral symbols was described by Agrippa von Nettesheim in his De occulta philosophia (1531) and more fully by Jan Bronkhorst of Nim- wegen in Holland who is named after his birthplace Noviomagus. 1 In 1539 he published at Cologne a tract, De numeris, in which he de- scribes numerals composed of straight lines or strokes which, he claims, were used by Chaldaei et Astrologi. Who these Chaldeans are whom he mentions it is difficult to ascertain; Cantor conjectures that they were late Roman or medieval astrologers. The symbols are given again in a document published by M. Host us in 1582 at Antwerp. An examina- tion of the symbols indicates that they enable one to write numbers up into the millions in a very concise form. But this conciseness is at- tained at a great sacrifice of simplicity; the burden on the memory is great. It does not appear as if these numerals grew by successive steps of time; it is more likely that they are the product of some in- ventor who hoped, perhaps, to see his symbols supersede the older (to him) crude and clumsy contrivances. An examination, in Figure 30, of the symbols for 1, 10, 100, and 1,000 indicates how the numerals are made up of straight lines. The same is seen in 4, 40, 400, and 4,000 or in 5, 50, 500, and 5,000. 98. General remarks. Evidently one of the earliest ways of re- cording the small numbers, from 1 to 5, was by writing the corre- sponding number of strokes or bars. To shorten the record in express- ing larger numbers new devices were employed, such as placing the bars representing higher values in a different position from the others, or the introduction of an altogether new symbol, to be associated with the primitive strokes on the additive, or multiplicative principle, or in some cases also on the subtractive principle. After the introduction of alphabets, and the observing of a fixed sequence in listing the letters of the alphabets, the use of these letters 1 See M. Cantor, Vorlesungen uber Geschichte der Mathematik, Vol. II (2d ed.; Leipzig, 1913), p. 410; M. Cantor, Mathemat. Beitrdge zum Kulturleben der Volker (Halle, 1863), p. 166, 167; G. Friedlein, Die Zahlzeichen und das elementare Rechnen der Griechen und Romer (Erlangen, 1869), p. 12; T. H. Martin, Annali di mate- maiica (B. Tortolini; Rome, 1863), Vol. V, p. 298; J. C. Heilbronner, Historia Mathcseos universae (Lipsiae, 1742), p. 735-37; J. Ruska, Archivfiir die Geschichte der Nalurwissenschaflen und Technik, Vol. IX (1922), p. 112-26. HINDU-ARABIC NUMERALS 69 for the designation of numbers was introduced among the Syrians, Greeks, Hebrews, and the early Arabs. The alphabetic numeral sys- tems called for only very primitive powers of invention; they made FIG. 30. The numerals described by Noviomagus in 1539. (Taken from J. C. Heilbronner, Historia malheseos [1742], p. 736.) 70 A HISTORY OF MATHEMATICAL NOTATIONS unnecessarily heavy demands on the memory and embodied no at- tempt to aid in the processes of computation. The highest powers of invention were displayed in the systems em- ploying the principle of local value. Instead of introducing new sym- bols for units of higher order, this principle cleverly utilized the posi- tion of one symbol relative to others, as the means of designating different orders. Three important systems utilized this principle: the Babylonian, the Maya, and the Hindu-Arabic systems. These three were based upon different scales, namely, 60, 20 (except in one step), and 10, respectively. The principle of local value applied to a scale with a small base affords magnificent adaptation to processes of computation. Comparing the processes of multiplication and division which we carry out in the Hindu-Arabic scale with_what tEe alpha- beticafsystems or the Roman system afforded places the superiority of the Hindu-Arabic scale in full view. The Greeks resorted to abacal computation, which is simply a primitive way of observing local value in computation. In what way the Maya or the Babylonians used their notations in computation is not evident from records that have come down to us. The scales of 20 or 60 would crJl for large multiplication tables. The orjgjn_and development of the Hindu-Arabic notation has received Intensive study. Nevertheless, little is known. An" outstand- ing facTis~^Iit~"cIuffng "the past one thousand years no uniformity in the shapes of the numerals has been reached. An American is some- times puzzled by the shape of the number 5 written in France. A European traveler in Turkey would find that what in Europe is a is in Turkey a 5. 99. Opinion of Laplace. Laplace 1 expresses his admiration for the invention of the Hindu-Arabic numerals and notation in this wise: "It is from the Indians that there has come to us the ingenious method of expressing all numbers, in ten characters, by giving them, at the same time, an absolute and a place value; an idea fine and important, which appears indeed so simple, that for this very reason we do not sufficiently recognize its merit. But this very simplicity, and the extreme facility which this method imparts to all calculation, place jour system of arithmetic in the first rank of the useful inventions. How difficult it was to invent such a method one can infer from the fact that it escaped the genius of Archimedes and of Apollonius of Perga, two of the greatest men of antiquity." 1 Exposition du systeme du monde (6th ed.; Paris, 1835), p. 376. Ill SYMBOLS IN ARITHMETIC AND ALGEBRA (ELEMENTARY PART) 100. In ancient Babylonian and Egyptian documents occur cer- tain ideograms and symbols which are not attributable to particular individuals and are omitted here for that reason. Among these signs is r~ for square root, occurring in a papyrus found at Kahun and now at University College, London, 1 and a pair of walking legs for squaring in the Moscow papyrus. 2 These symbols and ideograms will be referred to in our "Topical Survey" of notations. A. GROUPS OF SYMBOLS USED BY INDIVIDUAL WRITERS GREEK: DIOPHANTUS, THIRD CENTURY A.D. 101. The unknown number in algebra, defined by Diophantus as containing an undefined number of units, is represented by the Greek letter s with an accent, thus s', or in the form s'. In plural cases the symbol was doubled by the Byzantines and later writers, with the addition of case endings. Paul Tannery holds that the evidence is against supposing that Diophantus himself duplicated the sign. 3 G. H. F. Nesselmann 4 takes this symbol to be final sigma and remarks that probably its selection was prompted by the fact that it was the only letter in the Greek alphabet which was not used in writing num- bers. Heath favors "the assumption that the sign was a mere tachy- graphic abbreviation and not an algebraical symbol like our x, though discharging much the same function." 6 Tannery suggests that the sign is the ancient letter koppa, perhaps slightly modified. Other views on this topic are recorded by Heath. 1 Moritz Cantor, Vorlesungen uber Geschichte der Malhematik, Vol. I, 3d ed., Leipzig, p. 94. 2 B. Touraeff, Ancient Egypt (1917), p. 102. 3 Diophanti Alcxandrini opera omnia cum Graedst commentaries (Lipsiae, 1895), Vol. II, p. xxxiv-xlii; Sir Thomas L. Heath, Diophantus of Alexandria (2d ed.; Cambridge, 1910),. p. 32, 33. 4 Die Algebra der Griechen (Berlin, 1842), p. 290, 291. 5 Op. cit. y p. 34-36. 71 72 A HISTORY OF MATHEMATICAL NOTATIONS A square, z 2 , is in Diophantus' Arithmetica A F A cube, x 8 , is in Diophantus' Arithmetica K Y A square-square, z 4 , is in Diophantus' Arithmetica A r A A square-cube, z 5 , is in Diophantus' Arithmetica AK r A cube-cube, x 6 , is in Diophantus' Arithmetica K r K In place of the capital letters kappa and delta, small letters are some- times used. 1 Heath 2 comments on these symbols as follows: "There is no obvious connection between the symbol A y and the symbol s of which it is the square, as there is between x 2 and x, and in this lies the great inconvenience of the notation. But upon this notation no advance was made even by late editors, such as Xylander, or by Bachet and Fermat. They wrote N (which was short for Numerus) for the s of Diophantus, Q (Quadratus) for A F , C (Cubus) for K y , so that we find, for example, 1Q+5JV = 24, corresponding to z 2 +5z = 24. 3 Other symbols were however used even before the publication of Xylander's Diophantus, e.g., in Bombelli's Algebra" 102. Diophantus has no symbol for multiplication; he writes down the numerical results of multiplication without any preliminary step which would necessitate the use of a symbol. Addition is expressed 1 From Format's edition of Bachct ; s Diophantus (Toulouse, 1670), p. 2, Definition II, we quote: "Appellatvr igitur Quadratus, Dynamis, & est illius nota 5' superscriptum habens u sic S>. Qui autem sit ex quadrato in suum latus cubus est, cuius nota est \, superscriptum habens v hoc pacto w. Qui autem sit ex quad- rato in seipsum multiplicato, quadrato-quadratus est, cuius nota est geminum 5' habens superscriptum i>, hac ratione 55". Qui sit quadrato in cubum qui ab eodem latere profectus est, ducto, quadrato-cubus nominatur, nota eius 5/c superscriptum habens u sic 8i<y. Qui ex cubo in se ducto nascitur, cubocubus vocatur, & est eius nota geminum K superscriptum habens v, hoc pacto KK". Cui vero nulla harum proprietatum obtigit, sed constat multitudine vnitatem rationis experte, nurnerus vocatur, nota eius V Est et aliud signum immutabile definitorum, vnitas, cuius nota Jj. superscriptum habens 6 sic /z." The passage in Bachet's edition of 1621 is the same as this. 2 Op. tit., p. 38. 8 In Fermat's edition of Bachet's Diophantus (Toulouse, 1670), p. 3, Definition II, we read: "Haec ad verbum exprimenda esse arbitratus sum potius quam cum Xilandro nescio quid aliud comminisci. Quamuis enim in reliqua versione nostra notis ab eodem Xilandro excogitatis libenter vsus sim, quas tradam infra. Hie tamen ab ipso Diophanto longius recedere nolui, quod hac definitione notas ex- plicet quibus passim libris istis vtitur ad species omnes compcndio designandas, & qui has ignoret ne quidem Graeca Diophanti legere possit. Porr6 quadrat urn Dy- namin vocat, quae vox potestatem sonat, quia videlicet quadratus est veluti potestas cuius libet lineae, & passim ab Euclide, per id quod potest linea, quadratus illius designatur. Itali, Hispanique eadem ferd de causa Censum vocant, quasi INDIVIDUAL WRITERS 73 y mere juxtaposition. Thus the polynomial X 3 +13z 2 +5:r+2 would o _ o e in Diophantine symbols K F dA Y LyseMp, where M is used to repre- jnt units and shows that fi or 2 is the absolute term and not a part f the coefficient of s or x. It is to be noted that in Diophantus' square-cube" symbol for a; 5 , and "cube-cube" symbol for x 6 , the Iditive principle for exponents is employed, rather than the multipli- itive principle (found later widely prevalent among the Arabs and ^alians), according to which the "square-cube" power would mean x* id the "cube-cube" would mean # 9 . 103. Diophantus' symbol for subtraction is "an inverted ^ with le top shortened, A." Heath pertinently remarks: "As Diophantus scd no distinct sign for +, it is clearly necessary, in order to avoid mfusion, that all the negative terms in an expression, should be laced together after all the positive terms. And so in fact he does lace them." 1 As regards the origin of this sign /jv, Heath believes lat the explanation which is quoted above from the Diophantine ixt as we have it is not due to Diophantus himself, but is "an explana- on made by a scribe of a symbol which he did not understand." eath 2 advances the hypothesis that the symbol originated by placing I within the uncial form A> thus yielding A . Paul Tannery, 3 on the ;her hand, in 1895 thought that the sign in question was adapted om the old letter sampi !), but in 1904 he 4 concluded that it was ,ther a conventional abbreviation associated with the root of a cer- in Greek verb. His considerations involve questions of Greek gram- ar and were prompted by the appearance of the Diophantine sign ms rcdditum, prouentumque, qudd a latere seu radice, tanquam a feraci solo ladratus oriatur. Inde factum vt Gallorum nonnulli & Cermanorum corrupto cabulo zerizum appellarint. Numerum autem indeterminatum & ignotum, qui aliarum omnium potestatum latus esse intelligitur, Numerum sirnpliciter Dio- antus appellat. Alij passim Radicem, vel latus, vel rein dixerunt, Itali patrio cabulo Cosam. Caeterum nos in versione nostra his notis N. Q. C. QQ. QC. CC. signabimus Numerum, Quadratum, Cubum, Quadratoquadratum, Quadrato- bum, Cubocubum. Nam quod ad vnitates certas & determinatas spectat, eis tarn aliquam adscribere superuacaneum duxi, qu6d hae seipsis absque vlla ibiguitate sese satis indicent. Ecquis enim cum audit numerum 6. non statim ^itat sex vnitates? Quid ergo necesse est sex vnitates dicere, cum sufficiat dicere, c? . . . . " This passage is the same as in Bachet's edition of 1621. 1 Heath, op. cit., p. 42. 2 Ibid., p. 42, 43. 3 Tannery, op. cit., Vol. II, p. xli. 4 Bibliolheca mathematica (3d ser.), Vol. V, p. 5-8. 74 A HISTORY OF MATHEMATICAL NOTATIONS of subtraction in the critical notes to Schone's edition 1 of the Metrica of Heron. For equality the sign in the archetypal manuscripts seems to have been i\ "but copyists introduced a sign which was sometimes con- fused with the sign l|" (Heath). 104. The notation for division comes under the same head as the notation for fractions (see 41). In the case of unit fractions, a double accent is used with the denominator: thus y" = %. Sometimes a simple accent is used; sometimes it appears in a somewhat modified form as ^, or (as Tannery interprets it) as X ' thus y^~ J . For \- appear the symbols Z' and ^, the latter sometimes without the dot. Of fractions that are not unit fractions, f has a peculiar sign U7 of its own, as was the case in Egyptian notations. "Curiously enough," says Heath, "it occurs only four times in Diophantus." In some old manuscripts the denominator is written above the numerator, in some rare cases. Once we find ie 8 = * 4 5 , the denominator taking the position where we place exponents. Another alternative is to write the numerator first and the denominator after it in the same line, marking the denominator with a submultiple sign in some form : thus, =f y = | . 2 The following are examples of fractions from Diophantus : From v. 10: l ^ = ~ From v. 8, Lemma: 0ZV = 2 -J- J l * \- 8 V 250 From iv. 3: sX*/ = ~ From iv. 15: r ^ Fromvi. 12: ^M = (60z 2 +2,520)/(z 4 +900-60z 2 ) . 105. The fact that Diophantus had only one symbol for unknown quantity affected considerably his mode of exposition. Says Heath: "This limitation has made his procedure often very different from our modern work." As we have seen, Diophantus used but few symbols. Sometimes he ignored even these by describing an operation in words, when the symbol would have answered as well or better. Considering the amount of symbolism used, Diophantus' algebra may be desig- nated as "syncopated." 1 Heronis Alexandrini opera, Vol. Ill (Leipzig, 1903), p. 156, 1. 8, 10. The manuscript reading is novkbuv oSriS', the meaning of which is 74 jV 2 Heath, op. oil., p. 45, 47. INDIVIDUAL WRITERS 75 HINDU: BRAHMAGUPTA, SEVENTH CENTURY A.D. 106. We begin with a quotation from H. T. Colebrooke on Hindu algebraic notation: 1 "The Hindu algebraists use abbreviations and initials for symbols: they distinguish negative quantities by a dot, but have not any mark, besides the absence of the negative sign, to discriminate a positive quantity. No marks or symbols (other than abbreviations of words) indicating operations of addition or multipli- cation, etc., are employed by them: nor any announcing equality 2 or relative magnitude (greater or less) A fraction is indicated by placing the divisor under the dividend, but without a line of sepa- ration. The two sides of an equation are ordered in the same manner, one under the other The symbols of unknown quantity are not confined to a single one: but extend to ever so great a variety of denominations: and the characters used are the initial syllables of the names of colours, excepting the first, which is the initial of ydvat- tdvat, as much as." 107. In Brahmagupta, 3 and later Hindu writers, abbreviations occur which, when transliterated into our alphabet, are as follows: ru for rupa, the absolute number ya for ydvat-tdvat, the (first) unknown ca for calaca (black), a second unknown ni for nilaca (blue), a third unknown pi for pitaca (yellow), a fourth unknown pa for pandu (white), a fifth unknown lo for lohita (red), a sixth unknown c for caranij surd, or square root ya v for x 2 , the v being the contraction for varga, square number 108. In Brahmagupta, 4 the division of ru 3 c 450 c 75 c 54 by c 18 c 3 (i.e., 3+ V / 450+l / 75+1/54 by 1/18+1/3) is carried out as follows: "Put c 18 c 3. The dividend and divisor, multiplied by this, make ru 75 c 625. The dividend being then divided by the single surd ru 15 constituting the divisor, the quotient is ru 5 c 3." 1 H. T. Colebrooke, Algebra, with Arithmetic and Mensuration from the Sanscrit of Bramegupta and Bhdscara (London, 1817), p. x, xi. 2 The Bakhshali MS ( 109) was found after the time of Colebrooke and has an equality sign. *Ibid., p. 339 ff. 4 Brahme-sphuta-sidd'hdnta, chap. xii. Translated by H. T. Colebrooke in op. cit. (1817), p. 277-378; we quote from p. 342. A HISTORY OF MATHEMATICAL NOTATIONS | In modern symbols, the statement is, substantially: Multipl Mend and divisor by 1/18 1/3; the products are 75+1/675 an 15; divide the former by the latter, 5+1/3. "Question 16. 1 When does the residue of revolutions of the sur less one, fall, on a Wednesday, equal to the square root of two leg than the residue of revolutions, less one, multiplied by ten and aug mented by two? "The value of residue of revolutions is to be here put square c ydvat-tdvat with two added : ya v 1 ru 2 is the residue of revolutions Sanskrit character or letters, by which the Hindus denote the unknown quan- tities in their notation, are the following: TJJ,. offf, FIG. 31. Sanskrit symbols for unknowns. (From Charles Hutton, Mai In matical Tracts, II, 167.) The first symbol, pa, is the contraction for "white"; th second, ca, the initial for "black"; the third, ni, the initial for "blue"; the fourti pi, the initial for "yellow"; the fifth, lo, for "red." This less two isyavl; the square root of which isyal. Less one, it i ya 1 ru 1; which multiplied by ten is ya 10 ru 10; and augmented fy two, ya 10 ru 8. It is equal to the residue of revolutions yavl ru 2 less 1 a*. *. x f i xi i y a v ya 10 ru 8 ,, one; viz. yav I ru 1. Statement of both sides ' ~ ., . li/qua J ya v 1 ya ru 1 subtraction being made conformably to rule 1 there arises ya v I ya 10 Now, from the absolute number (9), multiplied by four times the [co efficient of the] square (36), and added to (100) the square of the [coefficient of the] middle term (making consequently 64), the square root being extracted (8), and lessened by the [coefficient of the] middle term (10), the remainder is 18 divided by twice the [coefficient of the square (2), yields the value of the middle term 9. Substituting with this in the expression put for the residue of revolutions, the answei comes out, residue of revolutions of the sun 83. Elapsed period ol days deduced from this, 393, must have the denominator in leasl terms added so often until it fall on Wednesday." 1 Colebrooke, op. ciL, p. 346. The abbreviations ru, c, ya, ya v, ca, ni, etc., an f raro1if rro f innu r\f fVr* /irki'r>c'r\/~kr/4irrr IriffoTO \r\ flo Sianalrrif alnVial-kftt INDIVIDUAL WRITERS 77 Notice that ya V ya J ? m J signifies Oz 2 +10z-8 = z 2 +0;r+l. 7/a v 1 ya ru 1 Brahmagupta gives 1 the following equation in three unknown quantities and the expression of one unknown in terms of the other two: "ya 197 ca 1644 nil ru ya ca ni ru 6302. Equal subtraction being made, the value of ydvat-tdvat is ca 1644 ni 1 ru 6302 ." M 197 In modern notation: whence, _1644?/+;g+6302 X ~~ ~ 197 HINDU: THE BAKHSHALI MS 109. The so-called Bakhshali MS, found in 1881 buried in the earth near the village of Bakhshali in the northwestern frontier of India, is an arithmetic written on leaves of birch-bark, but has come down in mutilated condition. It is an incomplete copy of an older manuscript, the copy having been prepared, probably about the eighth, ninth, or tenth century. "The system of notation/' says A. F. Rudolph Hoernle, 2 "is much the same as that employed in the arith- metical works of Brahmagupta and Bhaskara. There is, however, a very important exception. The sign for the negative quantity is a cross (+). It looks exactly like our modern sign for the positive quantity, but it is placed after the number which it qualifies. Thus 12 7 I means 127 (i.e. 5). This is a sign which I have not met with in any other Indian arithmetic ..... The sign now used is a dot placed over the number to which it refers. Here, therefore, there appears to be a mark of great antiquity. As to its origin I am not able to suggest any satisfactory explanation ..... A whole number, when it occurs in an arithmetical operation, as may be seen from the above given ex- ample, is indicated by placing the number 1 under it. This, however, is 1 Colebrooke, op. cit., p. 352. 2 "The Bakhshali Manuscript," Indian Antiquary, Vol. XVII (Bombay, 1888), p. 33-48, 275-79; see p. 34. 78 A HISTORY OF MATHEMATICAL NOTATIONS a practice which is still occasionally observed in India The following statement from the first example of the twenty-fifth siitra affords a good example of the system of notation employed in the Bakhshall arithmetic: 1 1 1 1 bhd 32 1 1 1 3+ 3+ 3+ phalarh 108 Here the initial dot is used much in the same way as we use the letter x to denote the unknown quantity, the value of which is sought. The number 1 under the dot is the sign of the whole (in this case, unknown) number. A fraction is denoted by placing one number under the other without any line of separation; thus Q is ,,, i.e. one-third. A mixed o o number is shown by placing the three numbers under one another; 1 ,1 1 thus 1 is 1+,, or 1 , i.e. one and one-third. Hence 1 3+ means 1 ,3 o i.e. - ). Multiplication is usually indicated by placing the numbers side by side; thus & 39. phalam 20 5 32 8 1 1 o means X32 = 20. Similarly 1 222 means .-;X.;X,-, or 333 8 3+ 3+ 3+ i.e. ~. Bhd is an abbreviation of bhdga, 'part/ and means that the number preceding it is to be treated as a denominator. Hence 111 8 27 111 bhd means 1 4- ~= or -^-. The whole statement, therefore, 3+ 3+ 3+ Z7 * 1 1 1 1 1 1 1 bhd 32 3+ 3+ 3+ phalam 108 , 27, means Q X 32 = 108, and may be thus explained, 'a certain number is o g found by dividing with ^ and multiplying with 32; that number is 108.' The dot is also used for another purpose, namely as one of the INDIVIDUAL WRITERS 79 ten fundamental figures of the decimal system of notation, or the zero (0123456789). It is still so used in India for both purposes, to indicate the unknown quantity as well as the naught The Indian dot, unlike our modern zero, is not properly a numerical figure at all. It is simply a sign to indicate an empty place or a hiatus. This is clearly shown by its name sdnya, 'empty/ .... Thus the two fig- ures 3 and 7, placed in juxtaposition (37), mean 'thirty-seven/ but with an 'empty space' interposed between them (3 7), they mean 'three hundred and seven/ To prevent misunderstanding the presence of the 'empty space' was indicated by a dot (3.7); or by what is now the zero (307). On the other hand, oc- v ; . ' currmg in the statement of a problem, the 'empty place 7 could be filled up, and here the -n . J0 ^ , l l .,, .,, , . ^' FIG. 32. From Bakhshah arithmetic dot which marked its presence (G . R . Kay6j Indian Mathematics [1915], signified a 'something' which p. 26; R. Hoernle, op. tit., p. 277). was to be discovered and to be put in the empty place In its double signification, which still survives in India, we can still discern an indication of that country as its birthplace The operation of multiplication alone is not indicated by any special sign. Addition is indicated by yu (for yuta), subtraction by + (ka for kanitaf) and division by bhd (for bhdga). The whole operation is commonly enclosed be- tween lines (or sometimes double lines), and the result is set down outside, introduced by pha (for phald)." Thus, pha served as a sign of equality. The problem solved in Figure 32 appears from the extant parts to have been : Of a certain quantity of goods, a merchant has to pay, as duty, , \, and J on three successive occasions. The total duty is 24. What was the original quantity of his goods? The solution ap- pears in the manuscript as follows: "Having subtracted the series from one," we get f , f , $ ; these multiplied together give | ; that again, subtracted from 1 gives f; with this, after having divided (i.e., in- verted, f), the total duty (24) is multiplied, giving 40; that is the original amount. Proof: multiplied by 40 gives 16 as the remainder. Hence the original amount is 40. Another proof: 40 multiplied by 1 and 1 } and 1 J gives the result 16; the deduction is 24; hence the total is 40. 80 A HISTORY OF MATHEMATICAL NOTATIONS HINDU: BHASKARA, TWELFTH CENTURY A.D. 110. Bhaskara speaks in his Lilavati 1 of squares and cubes of numbers and makes an allusion to the raising of numbers to higher powers than the cube. Ganesa, a sixteenth-century Indian commen- tator of Bhaskara, specifics some of them. Taking the words varga for square of a number, and g'hana for cube of a number (found in Bhas- kara and earlier writers), Ganesa explains 2 that the product of four like numbers is the square of a square, varga-varga; the product of six like numbers is the cube of a square, or square of a cube, varga-g'hana or g'hana-varga; the product of eight numbers gives varga-varga-varga; of nine, gives the cube of a cube, g'hana-g'hana. The fifth power was called varga-g'hana-ghdta; the seventh, varga-varga-g'hana-ghdta. 111. It is of importance to note that the higher powers of the unknown number are built up on the principle of involution, except the powers whose index is a prime number. According to this prin- ciple, indices are multiplied. Thus g'hana-varga does not mean n 3 -n 2 = n 5 , but (tt 3 ) 2 = n 6 . Similarly, g'hana-g'hana does not mean n 3 n 3 =n 6 , but (n 3 ) 3 = n 9 . In the case of indices that are prime, as in the fifth and seventh powers, the multiplicative principle became inoperative and the additive principle was resorted to. This is indicated by the word ghdta ("product")- Thus, varga-g'hana-ghdta means n 2 'n 3 = n 5 . In the application, whenever possible, of the multiplicative prin- ciple in building up a symbolism for the higher powers of a number, we see a departure from Diophantus. With Diophantus the symbol for x 2 , followed by the symbol for x 3 , meant x 5 ; with the Hindus it meant x 6 . We shall see that among the Arabs and the Europeans of the thirteenth to the seventeenth centuries, the practice was divided, some following the Hindu plan, others the plan of Diophantus. 112. In Bhaskara, when unlike colors (dissimilar unknown quanti- ties, like x and y) are multiplied together, the result is called bhavita ("product"), and is abbreviated bha. Says Colebrooke: "The prod- uct of two unknown quantities is denoted by three letters or syllables, as ya.ca bha, ca.ni bha, etc. Or, if one of the quantities be a higher power, more syllables or letters are requisite; for the square, cube, etc., are likewise denoted by the initial syllables, va, gha, va-va, va-gha, gha-gha,* etc. Thus ya va ca gha bha will signify the square of the 1 Colebrooke, op. cit., p. 9, 10. *Ibid., p. 10, n.3;p. 11. 8 Gha-gha for the sixth, instead of the ninth, power, indicates the use here of the additive principle. INDIVIDUAL WRITERS 81 first unknown quantity multiplied by the cube of the second. A dot is, in some copies of the text and its commentaries, interposed between the factors, without any special direction, however, for this notation." 1 Instead of ya va one finds in Brahmagupta and BhiLskara also the severer contraction ya v; similarly, one finds cav for the square of the second unknown. 2 It should be noted also that "equations are not ordered so as to put all the quantities positive; nor to give precedence to a positive term in a compound quantity: for the negative terms are retained, and even preferably put in the first place." 3 According to N. Raman ujacharia and G. R. Kaye, 4 the content of the part of the manuscript shown in Figure 33 is as follows: The *rtrv ,-* jSg^g^aS ^^^**^ S&iEHE^')^aft [b^Uf\lifr&%Jd&uJU^Ifc i^tih^^^g^t 8feUtfi%it / ' ._ ^*^kk^^^?i^^''^/6^>g FIG. 33. Sridhara's Trisdtika. Sridhara was born 991 A.D. Ho is cited by Bhaskara; he explains the "Hindu method of completing the square" in solving quadratic equations. circumference of a circle is equal to the square root of ten times the square of its diameter. The area is the square root of the product of ten with the square of half the diameter. Multiply the quantity whose square root cannot be found by any large number, take the square root of the product, leaving out of account the remainder. Divide it by the square root of the factor. To find the segment of a circle, take the sum of the chord and arrow, multiply it by the arrow, and square the product. Again multiply it by ten-ninths and extract its square root. Plane figures other than these areas should be calculated by considering them to be composed of quadrilaterals, segments of circles, etc. 1 Op. cit., p. 140, n. 2; p. 141. In this quotation we omitted, for simplicity, some of the accents found in Colebrooke's transliteration from the Sanskrit. 2 Ibid., p. 63, 140, 346. 3 Ibid., p. xii. 4 Bibliotheca mathematica (3d ser.), Vol. XIIT (1912-13), p. 206, 213, 214. 82 A HISTORY OF MATHEMATICAL NOTATIONS 113. Bhaskara Achdbrya, "Lilavati," 1 11 50 A D "Example: Tell me the fractions reduced to a common denominator which answer to three and a fifth, and one-third, proposed for addition; and those which correspond to a sixty-third and a fourteenth offered for sub- traction. Statement: 3 1 1 1 5 3 Answer: Reduced to a common denominator 45 3 5 G 53 15 15 15' bum i 5 - Statement of the second example: 1 1 63 14 ' Answer: The denominator being abridged, or reduced to least terms, by the common measure seven, the fractions become 1 1 9 2* Numerator and denominator, multiplied by the abridged denomina- p 2 9 tors, give respectively . ^p anc ^ i or Subtraction being made, the difference is . . 114. Bhaskara Achdbrya, "Vija-Ganita." 2 "Example: Tell quickly the result of the numbers three and four, negative or affirma- tive, taken together: .... The characters, denoting the quantities known and unknown, should be first written to indicate them gener- ally; and those, which become negative, should be then marked with a dot over them. Statement: 3 3*4. Adding them, the sum is found 7. Statement: 34. Adding them, the sum is 7. Statement: 3*4. Tak- ing the difference, the result of addition comes out 1. " 'So much as' and the colours 'black, blue, yellow and red/ 4 and others besides these, have been selected by venerable teaShers for names of values of unknown quantities, for the purpose of reckoning therewith. 1 Colebrooke, op. cit., p. 13, 14. 2 Ibid., p. 131. 3 In modern notation, 3+4 = 7, (-3) + (-4) = -7, 3 + (-4) = -1. 4 Colebrooke, op. cit., p. 139. INDIVIDUAL WRITERS 83 "Example: 1 Say quickly, friend, what will affirmative one un- known with one absolute, and affirmative pair unknown less eight absolute, make, if addition of the two sets take place? .... State- ment : 2 ya 1 ru 1 ya 2 ru 8 Answer: The sum is ya 3 ru 7. "When absolute number and colour (or letter) are multiplied one by the other, the product will be colour (or letter). When two, three or more homogeneous quantities are multiplied together, the product will be the square, cube or other [power] of the quantity. But, if unlike quantities be multiplied, the result is their (bhdvita) 'to be' product or factum. "23. Example: 3 Tell directly, learned sir, $he product of the multiplication of the unknown (ydvat-tdvat) five, less the absolute num- ber one, by the unknown (ydvat-tdvat) thrfce joined with the absolute two: .... Statement: 4 ya 5 ru 1 ^ , . . _ _ Product: ya v 15 ya 1 ru 2 . ya 3 ru 2 J u "Example: 5 'So much as' three, 'black' five, 'blue' seven, all affirmative: how many do they make with negative two, three, and one of the same respectively, added to or subtracted from them? Statement: 6 ya 3 ca 5 ni 7 Answer: Sum ya I ca 2 ni 6 . ya 2 ca 3 ni 1 Difference ya 5 ca 8 ni 8 . "Example: 7 Say, friend, [find] the sum and difference of two ir- rational numbers eight and two: .... after full consideration, if thou be acquainted with the sixfold rule of surds. Statement : 8 c 2 c 8. 1 Ibid. 2 In modern notation, x-\- 1 and 2x 8 have the sum 3x 7. 3 Colebrookc, op. cit., p. 141, 142. 4 In modern notation (5x - 1) (3x +2) = 15x 2 +7x -2. 5 Colcbrooke, op. cit., p. 144. 6 In modern symbols, 3x -\-5y-\-7z and 2x 3yz have the sum z-f 2y+Gz, and the difference 5z-f 8?/-f 8z. 7 Colebrooke, op. cit., p. 146. 8 In modern symbols, the example is l/8+/2 = vT8, T/8-1/^/2. The same example is given earlier by Brahmagupta in his Brahme-sputOrsidd'hdnta, chap, xviii, in Colebrooke, op. cit., p. 341. 84 A HISTORY OF MATHEMATICAL NOTATIONS Answer: Addition being made, the sum is c 18. Subtraction taking place, the difference is c 2." ARABIC: aL-KHOWARizMi, NINTH CENTURY A.D. 115. In 772 Indian astronomy became known to Arabic scholars. As regards algebra, the early Arabs failed to adopt either the Dio- phantine or the Hindu notations. The famous Algebra of al-Khowr- izmi of Bagdad was published in the original Arabic, together with an English translation, by Frederic Rosen, 1 in 1831. He used a manu- script preserved in the Bodleian Collection at Oxford. An examination of this text shows that the exposition was altogether rhetorical, i.e., devoid of all symbolism. "Numerals arc in the text of the work al- ways expressed by words: [Hindu-Arabic] figures are only used in some of the diagrams, arid in a few marginal notes." 2 As a specimen of al-Khowarizmi's exposition we quote the following from his Algebra, as translated by Rosen: "What must be the amount of a square, which, when twenty-one dirhems are added to it, becomes equal to the equivalent of ten roots of that square? Solution: Halve the number of the roots; the moiety is five. Multiply this by itself; the product is twenty-five. Subtract from this the twenty-one which are connected with the square; the remainder is four. Extract its root; it is two. Subtract this from the moiety of the roots, which is five; the remainder is three. This is the root of the square which you required, and the square is nine. Or you may add the root to the moiety of the roots; the sum is seven; this is the root of the square which you sought for, and the square itself is forty-nine." 3 By way of explanation, Rosen indicates the steps in this solution, expressed in modern symbols, as follows: Example: ARABIC: aL-KARKHi, EARLY ELEVENTH CENTURY A.D. 116. It is worthy of note that while Arabic algebraists usually build up the higher powers of the unknown quantit}' on the multiplica- tive principle of the Hindus, there is at least one Arabic writer, al- Karkhi of Bagdad, who followed the Diophantine additive principle. 4 1 The Algebra of Mohammed Ben Musa (cd. and trans. Frederic Rosen; London, 1831). See also L. C. Karpinski, Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi (1915). 2 Rosen, op. dt.., p. xv. 3 Ibid., p. 11. 4 See Cantor, op. cit., Vol. I (3d ed.), p. 767, 768; Heath, op. dt. t p. 41. INDIVIDUAL WRITERS 85 In al-Kharki's work, the Fakhri, the word mal means x 2 , ka c b means a 3 ; the higher powers are mal mal for x 4 , mdl ka c b for x 5 (not for x 6 ), ka L b ka c b for x 6 (not for x 9 ), wdZ moZ ka c b for x 7 (not for x 12 ), and so on. Cantor 1 points out that there are cases among Arabic writers where mdl is made to stand for x, instead of x 2 , and that this ambiguity is reflected in the early Latin translations from the Arabic, where the word census sometimes means x, and not x 2 . 2 BYZANTINE: MICHAEL PSELLUS, ELEVENTH CENTURY A.D. 117. Michael Psellus, a Byzantine writer of the eleventh century who among his contemporaries enjoyed the reputation of being the first of philosophers, wrote a letter 3 about Diophantus, in which he gives the names of the successive powers of the unknown, used in Egypt, which are of historical interest in connection with the names used some centuries later by Nicolas Chuquet and Luca Pacioli. In Psellus the successive powers are designated as the first number, the second n umber (square), etc. This nomenclature appears to have been borrowed, through the medium of the commentary by Hypatia, from Anatolius, a contemporary of Diophantus. 4 The association of the successive powers of the unknown with the series of natural numbers is perhaps a partial recognition of exponential values, for which there existed then, and for several centuries that followed Psellus, no ade- quate notation. The next power after the fourth, namely, x 5 , the Egyptians called "the first undescribed," because it is neither a square nor a cube; the sixth power they called the "cube-cube"; but the seventh was "the second undescribed," as being the product of the square and the "first undescribed." These expressions for x 6 and x 7 are closely related to Luca Pacioli's primo relato and secondo relato, found in his Summa of 1494. 5 Was Pacioli directly or indirectly in- fluenced by Michael Psellus? ARABIC: IBN ALBANNA, THIRTEENTH CENTURY A.D. 118. While the early Arabic algebras of the Orient are character- ized by almost complete absence of signs, certain later Arabic works on 1 Op. tit., p. 768. See also Karpinski, op. tit., p. 107, n. 1. 2 Such translations are printed by G. Libri, in his Histoire des stiences matht- matiques, Vol. I (Paris, 1838), p. 276, 277, 305. 3 Reproduced by Paul Tannery, op. tit., Vol. II (1895), p. 37-42. 4 See Heath, op. tit., p. 2, 18. 5 See ibid., p. 41; Cantor, op. tit., Vol. II (2d ed.), p. 317. 86 A HISTORY OF MATHEMATICAL NOTATIONS algebra, produced in the Occident, particularly that of al-Qalasadi of Granacla, exhibit considerable symbolism. In fact, as early as the thirteenth century symbolism began to appear; for example, a nota- tion for continued fractions in al-Ha$sar (391). Ibn Khaldun 1 states that Ibn Albanna at the close of the thirteenth century wrote a book when under the influence of the works of two predecessors, Ibn Almun c im and Alahdab. "He [Ibn Albanna] gave a summary of the demonstrations of these two works and of other things as well, con- cerning the technical employment of symbols 2 in the proofs, which serve at the same time in the abstract reasoning and the representa- tion to the eye, wherein lies the secret and essence of the explication of theorems of calculation with the aid of signs." This statement of Ibn Khaldun, from which it would seem that symbols were used by Arabic mathematicians before the thirteenth century, finds apparent confirmation in the translation of an Arabic text into Latin, effected by Gerard of Cremona (1114-87). This translation contains symbols for x and x 2 which we shall notice more fully later. It is, of course, quite possible that these notations were introduced into the text by the translator and did not occur in the original Arabic. As regards Ibn Albanna, many of his writings have been lost and none of his extant works contain algebraic symbolism. CHINESE: cnu SHIH-CHIEH (1303 A.D.) 119. Chu Shih-Chieh bears the distinction of having been "in- strumental in the advancement of the Chinese abacus algebra to the highest mark it has ever attained." 3 The Chinese notation is interest- ing as being decidedly unique. Chu Shih-Chieh published in 1303 a treatise, entitled Szu-yuen Yii-chien, or "The Precious Mirror of the Four Elements," from which our examples are taken. An expression like a+6+c+d, and its square, a 2 +6 2 +c 2 +d 2 +2a6+2ac+2ad+ 1 Consult F. Woepcke, "Rechcrches sur Fhistoire dcs sciences mathematiques chez les orientaux," Journal asiatique (5th ser.), Vol. IV (Paris, 1854), p. 369-72; Woepckc quotes the original Arabic and gives a translation in French. See also Cantor, op, cit., Vol. I (3d ed,), p. 805. 2 Or, perhaps, letters of the alphabet. 3 Yoshio Mikami, The Development of Mathematics in China and Japan (Leip- zig, 1912), p. 89. All our information relating to Chinese algebra is drawn from this book, p. 89-98. INDIVIDUAL WRITERS 87 2bc+2bd+2cd, were represented as shown in the following two illus- trations: 1 1 202 2 1*1 1 -X- 1 2 1 202 1 Where we have used the asterisk in the middle, the original has the character t'ai ("great extreme"). We may interpret this symbolism by considering a located one space to the right of the asterisk (#),& above, c to the left, and d below. In the symbolism for the square of a+b+c+d, the O's indicate that the terms a, 6, c, d do not occur in the expression. The squares of these letters are designated by the 1's two spaces from -K-. The four 2's farthest from -) stand for 2ab, 2ac, 2bc, 2bd, respectively, while the two 2's nearest to -)f stand for 2ac and 2bd. One is impressed both by the beautiful symmetry and by the extreme limitations of this notation. 120. Previous to Chu Shih-Chieh's time algebraic equations of only one unknown number were considered ; Chu extended the process to as many as four unknowns. These unknowns or elements were called the "elements of heaven, earth, man, and thing." Mikami states that, of these, the heaven clement was arranged below the known quantity (which was called "the great extreme"), the earth clement to the left, the man element to the right, and the thing ele- ment above. Letting -)f stand for the great extreme, and x, y, z, u, for heaven, earth, man, thing, respectively, the idea is made plain by the following representations : Mikami gives additional illustrations: 1 1 0- 2 # 1 1 +2yz xz+z 2 88 A HISTORY OF MATHEMATICAL NOTATIONS Using the Hindu-Arabic numerals in place of the Chinese calculating pieces or rods, Mikami represents three equations, used by Chu, in the following manner: In our notation, the four equations are, respectively, a) 2x 6) z 2 + i/ 2 - 2 2 = 0, c) 2x +2y - u = 0. No sign of equality is used here. All terms appear on one side of the equation. Notwithstanding the two-dimensional character of the notation, which permits symbols to be placed above and below the starting-point, as well as to left and right, it made insufficient pro- vision for the representation of complicated expressions and for easy methods of computation. The scheme does not lend itself easily to varying algebraic forms. It is difficult to see how, in such a system, the science of algebra could experience a rapid and extended growth. The fact that Chinese algebra reached a standstill after the thirteenth century may be largely due to its inelastic and faulty notation. BYZANTINE: MAXIMUS PLANUDES, FOURTEENTH CENTURY A.D. 121. Maximus Planudes, a monk of the first half of the fourteenth century residing in Constantinople, brought out among his various compilations in Greek an arithmetic, 1 and also scholia to the first two books of Diophantus' Arithmetical These scholia are of interest to us, for, while Diophantus evidently wrote his equations in the running text and did not assign each equation a separate line, we find in Planudes the algebraic work broken up so that each step or each equation is assigned a separate line, in a manner closely resembling modern practice. To illustrate this, take the problem in Diophantus (i. 29), 1 Das Recheribuch des Maximus Planudes (Halle: herausgegeben von C. I. Gcrhardt, 1865). 2 First printed in Xylander's Latin translation of Diophantus' Arilhmetica (Basel, 1575). These scholia in Diophantus are again reprinted in P. Tannery, Diophanti Alexandrini opera omnia (Lipsiae, 1895), Vol. II, p. 123-255; the ex- ample which we quote is from p. 201. INDIVIDUAL WRITERS 89 "to find two numbers such that their sum and the difference of their squares are given numbers." We give the exposition of Planudes and its translation. Planudes Translation K TT [Given the numbers], 20, 80 eK& sd/xZ M *Asa Putting for the numbers, x+10, 10-z rerp A F dss/c^p A Y d{jLp A ss/c . . .Squaring, 2 +20#+100, z 2 +100-20z virepox* ssju l ff " JUTT Taking the difference, 40# = 80 /xep sd I 9 * jjpp Dividing, x = 2 for M^ AM?' Result, 12, 8 ITALIAN: LEONARDO OF PISA (1202 A.D.) 122. Leonardo of Pisa's mathematical writings are almost wholly rhetorical in mode of exposition. In his Liber abbaci (1202) he used the Hindu-Arabic numerals. To a modern reader it looks odd to see expressions like -$ "A t 42, the fractions written before the integer in the case of a mixed number. Yet that mode of writing is his invariable practice. Similarly, the coefficient of x is written after the name for x, as, for example, 1 "radices ^12" for I2%x. A computation is indi- cated, or partly carried out, on the margin of the page, and is inclosed in a rectangle, or some irregular polygon whose angles are right angles. The reason for the inverted order of writing coefficients or of mixed numbers is due, doubtless, to the habit formed from the study of Arabic works; the Arabic script proceeds from right to left. Influ- enced again by Arabic authors, Leonardo uses frequent geometric figures, consisting of lines, triangles, and rectangles to illustrate his arithmetic or algebraic operations. He showed a partiality for unit fractions; he separated the numerator of a fraction from its denomi- nator by a fractional line, but was probably not the first to do this ( 235). The product of a and b is indicated by factus ex.a.b. It has boon stated that he denoted multiplication by juxtaposition, 2 but G. Enestrom shows by numerous quotations from the Liber abbaci that such is not the case. 3 Cantor's quotation from the Liber abbaci, 1 II liber abbaci di Leonardo Pisano (ed. B. Boncornpagni), Vol. I (Rome, 1867), p. 407. 2 Cantor, op. tit., Vol. II (2d ed.), p. 62. 3 BMiotheca mathematica (3d ser.), Vol. XII (1910-11), p. 335, 336. 90 A HISTORY OF MATHEMATICAL NOTATIONS "sit numerus .a.e.c. quaedam coniunctio quae uocetur prima, numeri vero .d.b.f. sit coniunctio secunda," 1 is interpreted by him as a product, the word coniunctio being taken to mean "product." On the other hand, Kncstrom conjectures that numerus should be numeri, arid trans- lates the passage as meaning, "Let the numbers a, e, c be the first, the numbers d, 6, / the second combination." If Enestrom's interpreta- tion is correct, then a.e.c and d.b.f are not products. Leonardo used in his Liber abbaci the word res for x, as well as the word radix. Thus, he speaks, "et intellige pro re summam aliquarn ignotam, quarn inuenire uis." 2 The following passage from the Liber abbaci contains the words numerus (for a given number), radix for x, and census for x 2 : "Primus cniin modus est, quando census et radices equantur numero. .... Verbi gratia: duo census, et decem radices equantur denariis 30," 3 i.e., 2x' 2 +10x = 30. The use of res for x is found also in a Latin translation of al-Khowarizmi's algebra, 4 due perhaps to Gerard of Cremona, where we find, "res in rein fit census/' i.e., x.x = x 2 . The word radix for x as well as res, and substantia for a; 2 , are found in Robert of Chester's Latin translation of al-Khowarizmi's algebra. 5 Leonardo of Pisa calls x 3 cubus, x* census census, X Q cubus cubus, or else census census census; he says, " . . . . est multiplicare per cubum cubi, sicut multiplicare per censum census census." 6 He goes even farther and lets x 8 be census census census census. Observe that this phrase- ology is based on the additive principle x 2 y? x 2 x 2 = x s . Leonardo speaks also of radix census census. 7 The first appearance of the abbreviation R or H for radix is in his Pradica geometriae (1220), 8 where one finds the R meaning "square root" in an expression "et minus I}. 78125 dragme, et diminuta radice 28125 dragme." A few years later, in Leonardo's Flos, 9 one finds marginal notes which are abbreviations of passages in the text relating to square root, as follows: 1 Op. cit., Vol. I (3d ed.), p. 132. 2 Ibid., Vol. I, p. 191. 3 Ibid., Vol. I, p. 407. 4 Libri, Histmre des sciences mathematiques en Italic, Vol. I (Paris, 1838), p. 268. 6 L. C. Karpinski, op. cit., p. 68, 82. 6 II liber abbaci, Vol. I, p. 447. 7 Ibid., Vol. I, p. 448. 8 Scritti di Leonardo Pisano (ed. B. Boncompagni), Vol. II (Rome, 1862), p. 209. 9 Op. cit., Vol. II, p. 231. For further particulars of the notations of Leonardo of Pisa, see our 219, 220, 235, 271-73, 290, 292, 318, Vol. II, 389. INDIVIDUAL WRITERS 91 .R. x p { . Bino. ij for primi [quidem] binomij radix 2 . ' B . R . x for radix [quippe] secundi binomij .Bi. 3 { . R. x for Tertij [autem] binomij radix .Bi. 4 l . R. x for Quarti [quoque] binomij radix FRENCH: NICOLE ORESME, FOURTEENTH CENTURY A.D. 123. Nicole Oresme (ca. 1323-82), a bishop in Normandy, pre- pared a manuscript entitled Algorismus proportionum, of which several copies arc extant. 1 He was the first to conceive the notion of fractional powers which was afterward rediscovered by Stevin. More than this, he suggested a notation for fractional powers. He considers powers of ratios (called by him proportiones) . Representing, as does Oresme himself, the ratio 2:1 by 2, Oresme expresses 2* by the symbolism 1 v - ~ and reads this medietas [proportionis] duplae; he expresses (20* by the symbolism l.p.l 4.2.2 and reads it quarta pars [proportionis] duplae sesquialterae. The fractional exponents { and | are placed to the left of the ratios affected. H. Wicleitner adds that Oresme did not use these symbols in com- putation. Thus, Oresme expresses in words, ". . . . proponatur pro- portio, que sit due tertie quadruple; et quia duo est numerator, ipsa erit vna tertia quadruple duplicate, sev sedecuple," 2 i.e., 4 = (4 2 )* = 16*. Oresme writes 3 also: "Sequitur quod .a. moueatur velocius .6. in pro- portione, que est medietas proportionis .50. ad .49.," which means, "the velocity of a -.velocity of 6 = 1/50:1/49," the word medietas mean- ing "square root." 4 The transcription of the passage shown in Figure 34 is as follows: ''Una media debet sic scribi una tertia sic et due tertie sic et sic de alijs. et numerus, qui supra uirgularn, dicitur 1 Maximilian Curtze brought out an edition after the MS R. 4 2 of the Gyrn- nasiat-Bihliothck at Thorn, under the title Der Algorithmic Proportionum des Nicolaus Oresme (Berlin, 1868). Our photographic illustration is taken from that publication. 2 Curtze, op. cit., p. 15. 3 Ibid., p. 24. 4 See Knestrom, op. tit., Vol. XII (1911-12), p. 181. For further details see also Curtze, Zeitschrift fur Mathematik und Phijsik, Vol. XIII (Suppl. 18G8), p. 65 flf. 92 A HISTORY OF MATHEMATICAL NOTATIONS numerator, iste uero, qui est sub uirgula, dicitur denominator. 2. Pro- portio dupla scribitur isto modo 2. Za , et tripla isto modo 3. Za ; et sic de alijs. Proportio sesquialtera sic scribitur , et sesquitertia 1 3 . Proportio superpartiens duas tertias scribitur sic P 2 13. Proportio dupla superpartiens duas quartas scribitur sic P 2 24 ;et sic de alijs. 3. Medietas duple scribitur sic Ip 2-2 , quarta pars duple sesquialtere scribitur sic 4-2-2 ; et sic de alijs." FIG. 34. From the first page of Oresme's Algorismus proportionum (four- teenth century). A free translation is as follows: "Let a half be written , a third ~ , and two-thirds and so on. And the number above the line is called the 'numerator/ the one below the line is called the 'denominator. 1 2. A double ratio is written in this manner 2. Zo , a triple in this manner 3.', and thus in other cases. The ratio one and one-half is written , and one and one-third is written . The ratio one and two-thirds is written . A double ratio and two-fourths are written , and thus INDIVIDUAL WRITERS 93 in other cases. 3. The square root of two is written thus l.p 2 2 , the thus , and fnnrfli rnof" rvf fwn pnH nnp-Vnlf is wriffpn fHiis 1 - * 1U HI I'll ivJvfti \JL \j\\\J clillvl \JllC/HCvll ID Wllvvdl 1/llLlo I . ^ -. 1 4. 2. 2i in other cases." ARABIC: AL-QALASAD!, FIFTEENTH CENTURY A.D. 124. Al-Qalasadi's Raising of the Veil of the Science of Gubar ap- peared too late to influence the progress of mathematics on the European Continent. Al-Qalasadf used -^, the initial letter in the Arabic word jidr, "square root"; the symbol was written above the number whose square root was required and was usually separated from it by a horizontal line. The same symbol, probably considered this time as the first letter mjahala ("unknown"), was used to repre- sent the unknown term in a proportion, the terms being separated by the sign .'. . But in the part of al-Qalasadi r s book dealing more par- ticularly with algebra, the unknown quantity x is represented by the letter \Jp, x 2 by the letter \jo, x 3 by the letter $"; these are written above their respective coefficients. Addition is indicated by juxta- position. Subtraction is *tfj ; the equality sign, J , is seen to resem- ble the Diophantine t, if we bear in mind that the Arabs wrote from right to left, so that the curved stroke faced in both cases the second member of the equation. We reproduce from Woepcke's article a few samples of al-Qalasadi's notation. Observe the peculiar shapes of the Hindu- Arabic numerals (Fig. 35). Woepcke 1 reproduces also symbols from an anonymous Arabic manuscript of unknown date which uses symbols for the powers of x and for the powers of the reciprocal of x, built up on the additive prin- ciple of Diophantus. The total absence of data relating to this manu- script diminishes its historic value. GERMAN: REGIOMONTANUS (ca. 1473) 125. Regiomontanus died, in the prime of life, in 1476. After having studied in Rome, he prepared an edition of Ptolemy 2 which was issued in 1543 as a posthumous publication. It is almost purely rhetorical, as appears from the following quotation on pages 21 and 22. 1 Op. cit., p. 375-80. 2 loannis de Monte Regio et Georgii Pvrbachii epitome, in Cl. Ptolemaei magnam compositionem (Basel, 1543). The copy examined belongs to Mr. F. E. Brasch. 94 A HISTORY OF MATHEMATICAL NOTATIONS By the aid of a quadrant is determined the angular elevation ACE, "que erit altitude tropici hiemalis," and the angular elevation ACF y "que erit altitudo tropici aestivalis," it being required to find the arc EF between the two. "Arcus itaque EF y fiet distantia duorum tropi- 9 & 1 ... ^; \/54... ^; j FORMULES D^QUATIONS TRINOMES. 2. (ft, PROPORTIONS. 7 : 12 = 84 : x ..... ^ . ii ; 20 == 66 : ...... a^. A 6:6 .-. &>o A || . Fir,. 35. Al-Qalasddi's algebraic symbols. (Compiled by F. Woepcko, Journal asialique [Oct. and Nov., 1854], p. 363, 364, 366.) corum quesita. Hac Ptolemaeus reperit 47. graduum 42. minutorum 40. secundorum. Inuenit enim proportionem eius ad totum circulu sicut 11. ad 83, postea uero minorem inuenerunt. Nos autem inuenimus arcum AF 65. graduum 6. minutorum, & arcum AE 18. graduum 10. INDIVIDUAL WRITERS 95 minutorum. Ideoq. nunc distantia tropicorum est 46. graduum 56. niinutorum, ergo declinatio soils maxima nostro tempore est 23. graduum 28. minutorum. " 126. We know, however, that in some of his letters and manu- scripts symbols appear. They are found in letters and sheets contain- ing computations, written by Regiomontanus to Giovanni Bianchini, Jacob von Speier, and Christian Roder, in the period 1463-71. These documents are kept in the Stadtbibliothek of the city of Niirnberg. 1 Regiomontanus and Bianchini designate angles thus: gr 35 m 17; Regiomontanus writes also: 44. 42'. 4" (see also 127). In one place 2 Regiomontanus solves the problem: Divide 100 by a certain number, then divide 100 by that number increased by 8; the sum of the quotients is 40. Find the first divisor. Regiomontanus writes the solution thus : In Modern Symbols "100 _JP9_ 100 JLOO \TJe l^et'8 x x+8 100^ et 800 100z+800 800 200S+800 sf+Sx 40c et 320^ 200^ et 800 40x 2 +320x = 200^+800. 40c et 120 if 800 40x 2 + I20x = 800 IcCet 3^f 20 x 2 + 3z = 20 I J- addo numerum 20!J 8 /- f 1 !} add the no. 20J- = * Radix quadrata de - 8 4 9 minus | 1 ^ l^ 8 / * = x Primus ergo divisor fuit 1} de 22 j Hence the first divisor was I? 11." 1/221 -H. Note that "plus" is indicated here by et; "minus" by 19, which is probably a ligature or abbreviation of "minus." The unknown quan- tity is represented by ^ and its square by c- Besides, he had a sign for equality, namely, a horizontal dash, such as was used later in Italy by Luca Pacioli, Ghaligai, and others. See also Fig. 36. 1 Curtze, Urkunden zur Gcschichte der Mathemalik im Mittelalter und der Re- naissance (Leipzig, 1902), p. 185-336 = A bhandlungen zur Geschichle der Mathe- matik, Vol. XII. Sec -also L. C. Karpinski, Robert of Chester's Translation of the Algebra of Al-Khowarizmi (1915), p. 36, 37. 2 Curtze, op. tit., p. 278. 96 A HISTORY OF MATHEMATICAL NOTATIONS 127. Figure 37 1 illustrates part of the first page of a calendar issued by Regiomontanus. It has the heading Janer (" January "). Farther to the right are the words Sunne Monde Stainpock ("Sun Moon Capricorn")- The first line is 1 A. Kl. New Jar (i.e., "first day, A. calendar, New Year"). The second line is 2. b. 4. no. der achtet S. Stephans. The seven letters A , 6, c, d y e, F, g, in the second column on the left, are the dominical letters of the calendars. Then come the days of the Roman calendar. After the column of saints' days comes a double column for the place of the sun. Then follow two double columns for the moon's longitude; one for the mean, the other for the 10 -x 100- 10* s*- Wx % x* + 101) - 20x FIG. 36. Computations of Regiomontanus, in letters of about 1460. (From manuscript, Niirnberg, fol. 23. (Taken from J. Tropfke, Geschichte der Elementar- Mathematik (2d ed.), Vol. II [1921], p. 14.) true. The S signifies signum (i.e., 30); the G signifies gradus, or "degree." The numerals, says De Morgan, are those facsimiles of the numerals used in manuscripts which are totally abandoned before the end of the fifteenth century, except perhaps in reprints. Note the shapes of the 5 and 7. This almanac of Regiomontanus and the Compotus of Anianus are the earliest almanacs that appeared in print. ITALIAN: THE EARLIEST PRINTED ARITHMETIC (1478) 128. The earliest arithmetic was printed anonymously at Treviso, a town in Northeastern Italy. Figure 38 displays the method of solv- ing proportions. The problem solved is as follows: A courier travels from Rome to Venice in 7 days; another courier starts at the same time and travels from Venice to Rome in 9 days. The distance be- tween Rome and Venice is 250 miles. In how many days will the 1 Reproduced from Karl Falkenstein, Geschichte der Buchdruckerkunst (Leip- zig, 1840), Plate XXIV, between p. 54 and 55. A description of the almanac of Regiomontanus is given by A. de Morgan in the Companion to the British Almanac, for 1846, in the article, "On the Earliest Printed Almanacs," p. 18-25. INDIVIDUAL WRITERS 97 couriers meet, and how many miles will each travel before meeting? Near the top of Figure 38 is given the addition of 7 and 9, and the <<tlettto* fce SPfttgifte* 3of><ttttt toott Uttn^erf * ( Johannes Regioinontanas. ) FIG. 37. "Calendar des Magister Johann von Kunspcrk (Johannes Reio- montanus) Nilrnberg um 1473." division of 63 by 16, by the scratch method. 1 The number of days is 3-H-. The distance traveled by the first courier is found by the pro- 1 Our photograph IB taken from the Alii dell'Accademia Pontificia de' nuovi Lincei, Vol. XVI (Roma, 1863), p. 570. 98 A HISTORY OF MATHEMATICAL NOTATIONS portion 7:250 = ff:x. The mode of solution is interesting. The 7 and 250 are written in the form of fractions. The two lines which cross e qudiite migli* baucra fatto citifcluduno t lose. if a fe f ondo la ritgula cofu , i 6 partitoje r^ 50201 1 6 Sc tu t? uol fap^tt quaita tm$lia bauera fatto ci& cbjdunoifa per b neguU'oel.^'Dicendo ptimo per qucllui t>a T\oma. i i -t * ^^ ^ 5 o ' T" X T it) * *fo -* 1 *5o f o o 5T-< roji **\ *X*S \i 40 ^^tf ^-f-f < 5 f of ^. * f ^ /QuelhiKbe vuntoa rsoma bauera tatto tr .1 4 o.e __ poi mettila riegula per 7 d co:nn-o tea Uene^ua. 44 FIG. 38. From the earliest printed arithmetic, 1478 and the two horizontal lines on the right, connecting the two numer- ators and the two denominators, respectively, indicate what numbers INDIVIDUAL WRITERS 99 are to be multiplied together: 7X1X16 = 112; 1X250X63 = 15,750. The multiplication of 250 and 63 is given; also the division of 15,750 *>i it tttiwti JaJTccbe fa.t *}$.Hi pcrluwq? modi qui fottofcritri. 4 <*|4 * * 7 i f6\ i 7 o 3 4 7 [ a. -* ^ 7 ^ 9 X* I i J 5 T 5' / ^ua-7 001^6^6) ' 5 ^ 7 y 9 } fc?uma. 7 o o t t 6 * 2j^ o 3 67, i 3 5 '8 /"L 3 o o ~ffi * / FIG. 39. Multiplications in the Treviso arithmetic; four multiplications of 56,789 by 1,234 as given on one page of the arithmetic. 100 A HISTORY OF MATHEMATICAL NOTATIONS by 112, according to the scratch method. Similarly is solved the proportion 9 : 250 = J jf : x. Notice that the figure 1 is dotted in the same way as the Roman I is frequently dotted. Figure 39 represents other examples of multiplication. 1 FRENCH: NICOLAS CHUQUET (1484) 129. Over a century after Oresme, another manuscript of even greater originality in matters of algebraic notation was prepared in France, namely, Le triparly en la science des nombres (1484), by Nicolas Chuquet, a physician in Lyons. 2 There are no indications that he had seen Oresrne's manuscripts. Unlike Oresme, he does not use fractional exponents, but he has a notation involving integral, zero, and negative exponents. The only possible suggestion for such exponential notation known to us might have come to Chuquet from the Gobar numerals, the Fihrist, and from the scholia of Neophytos ( 87, 88) which are preserved in manuscript in the National Library at Paris. Whether such connection actually existed we are not able to state. In any case, Chuquet elaborates the exponential notation to a completeness apparently never before dreamed of. On this sub- ject Chuquet was about one hundred and fifty years ahead of his time; had his work been printed at the time when it was written, it would, no doubt, have greatly accelerated the progress of algebra. As it was, his name was known to few mathematicians of his time. Under the head of "Numeration," the Triparly gives the Hindu- Arabic numerals in the inverted order usual with the Arabs: ".0.9.8.7.6.5.4.3.2.1." arid included within dots, as was customary in late manuscripts and in early printed books. Chuquet proves addition by "casting out the 9's," arranging the figures as follows: 5 2 >.7. ^ 16 9. ' .7. 1 Ibid., p. 550. 2 Op. cit. (public d'apres le manuscrit fonds Francois N. 1346 de la Biblio- theque nationale de Paris et precede .(Time notice, par M. Aristide Marre), Bullettino di Bibliog. e di Storia delle scienze mat. etfisiche, Vol. XIII (1880), p. 555- 659, 693-814; Vol. XIV, p. 413-60. INDIVIDUAL WRITERS 101 The addition of $ and f is explained in the text, and the following arrangement of the work is set down by itself: 1 130. In treating of roots he introduces the symbol R, the first letter in the French word ratine and in the Latin radix. A number, say 12, he calls ratine premiere, because 12, taken as a factor once, gives 12; 4 is a ratine seconde of 16, because 4, taken twice as a factor, gives 16. He uses the notations #M2. equal .12., $ 2 .16. equal .4., /? 4 .16. equal .2., # 5 .32. equal .2. To quote: "II conuiendroit dire que racine piniere est entenduc pour tous nombres simples Come qui diroit la racine premiere de .12. que Ion peult ainsi noter en mettant .1. dessus R. en ceste maniere R l .12. cest .12. Et #*.9. est .9. et ainsi de tous aultres nobres. Racine seconde est celle qui posee en deux places lune soubz laultre et puys multipliee lune par laultre pduyt le nombre duquel elle est racine seconde Comme 4. et .4. qui multipliez lung par laultre sont .16. ainsi la racine seconde de .16. si est .4. ... on le peult ainsi rnettre # 2 16. ... Et $ 5 .32. si est .2. Racine six! se doit ainsi mettrc I}, et racine septiesrne ainsi I} 7 . ... Aultres maniercs de racines sont que les simples devant dictes que Ion peult appeller racines composees Come de 14. plus # 2 180. dont sa racine seconde si est .3. p. # 2 5. [i.e., 1/14+ 1/180 = 3+ 1/5] ... coe la racine seconde de .14. p # 2 .180. se peult ainsi rnettre g 2 .14.p.g 2 .18Q." 2 Not only have we here a well-developed notation for roots of inte- gers, but we have also the horizontal line, drawn underneath the binomial 14+ 'V 180, to indicate aggregation, i.e., to show that the square root of the entire binomial is intended. Chuquet took a position in advance of his time when he computed with zero as if it were an actual quantity. He obtains, 3 according to his rule, z = 2 1/4^-4 as the roots of 3x 2 +12-12x. He adds: "... reste .0. Done 5 2 .0. adioustee ou soustraicte avec .2. ou de .2. monte .2. qui est le nob? que Ion demande." 131. Chuquet uses p and ra to designate the words plus and mains. These abbreviations we shall encounter among Italian writers. Pro- ceeding to the development of his exponential theory and notation, 1 Boncomp,gni, Bullettino, Vol. XIII, p. 636. 9 Ibid. t p. 655. 3 Ibid., p. 805; Enestrom, Bibliotheca mathematica, Vol. VIII (1907-8), p. 203. 102 A HISTORY OF MATHEMATICAL NOTATIONS he states first that a number may be considered from different points of view. 1 One is to take it without any denomination (sans aulcune denomiaciori) , or as having the denomination 0, and mark it, say, .12? and .13? Next a number may be considered the primary number of a continuous quantity, called "linear number" (nombre linear), des- ignated .12 1 .13 1 .20 1 , etc. Third, it may be a secondary or superficial number, such as 12 2 . 13 2 . 19 2 ., etc. Fourth, it may be a cubical num- ber, such as .12 3 . 15 3 . I 3 ., etc. "On les peult aussi entendre estre nombres quartz ou quarrez de quarrez qui seront ainsi signez . 12 4 . 18 4 . 30 4 ., etc." This nomenclature resembles that of the Byzantine monk Psellus of the eleventh century ( 117). Chuquet states that the ancients called his primary numbers "things" (choses) and marked them .P.; the secondary numbers they called "hundreds" and marked them .tf.; the cubical numbers they indicated by D ; the fourth they called "hundreds of hundreds" (champs de champ), for which the character was ttf. This ancient nomenclature and notation he finds insufficient. He introduces a symbolism "que Ion peult noter en ceste maniere # 2 .12 l . # 2 .12 2 . $ 2 .12 3 . # 2 .12 4 . etc. S 3 .!^. 5 3 .12 2 . U2. # 3 .12 4 . etc. # 4 .13 5 . #<M2 6 . etc." Here "7i 4 .13 5 ." means l/13x 5 . He proceeds further and points out "que Ion peult ainsi noter .12! ' m ' ou moins 12.," thereby introducing the notion of an exponent "minus one." As an alternative notation for this last he gives ".rft.121," which, however, is not used again in this sense, but is given another interpretation in what follows. From what has been given thus far, the modern reader will prob- ably be in doubt as to what the symbolism given above really means. Chuquet's reference to the ancient names for the unknown and the square of the unknown may have suggested the significance that he gave to his symbols. His 12 2 does not mean 12X12, but our 12x 2 ; the exponent is written without its base. Accordingly, his ".12. 1 --" means 12X" 1 . This appears the more clearly when he comes to "adi- ouster 8! avec m.5! monte tout .3. 1 Ou .10. 1 avec .ra.16. 1 mote tout m.6. 1 ," i.e., 8x-5x = 3x, 10o:--16x= -6z. Again, ".8. 2 avec .12. 2 montent .20. 2 " means 8x 2 +12x 2 = 20x 2 ; subtracting ".w.16?" from ".12. 2 " leaves "12. 2 m. m. 169 qui valent autant c5me .12. 2 p. 16?" 2 The meaning of Chuquet's ".12?" appears from his "Example, qui multiplie .12? par .12? montent .144. puis qui adiouste .0. avec .0. monte 0. ainsi monte ceste multiplicacion .144?," 3 i.e., 12xX12o: = 1 Boncompagni, op. cit., Vol. XIII, p. 737. * Ibid., p. 739. 3 Ibid., p. 740. INDIVIDUAL WRITERS 103 144x. Evidently, =1; he has the correct interpretation of the ex- ponent zero. He multiplies .12? by .10. 2 and obtains 120. 2 ; also .5. 1 times .8. 1 yields .40. 2 ; .12. 3 times .10. 5 gives .120. 8 ; .8. 1 times .7 1 -- gives .56? or .56.; .8? times .7 1 -- gives .56. 2 Evidently algebraic multiplication, involving the product of the coefficients and the sum of the exponents, is a familiar process with Chuquet. Nevertheless, he does not, in his notation, apply exponents to given numbers, i.e., with him "3 2 " never means 9, it always means 3z 2 . He indicates (p. 745) the division of 30 x by x*+x in the following manner: 30. m. I 1 I 2 p. I 1 ' As a further illustration, we give # 2 y.g# 2 24.p.# 2 14. multiplied by # 2 lj^#?_24. ra#?14. gives & 24. This is really more compact and easier to print than our J/1I+1/24+1/1-J- times V\\ equals 1/24 . FRENCH: STIENNE DE LA ROCHE (1520) 132. Estienne de la Roche, Villefranche, published Larismethique, at Lyon in 1520, which appeared again in a second edition at Lyon in 1538, under the revision of Gilles Huguetan. De la Roche mentions Chuquet in two passages, but really appropriates a great deal from his distinguished predecessor, without, however, fully entering into his spirit and adequately comprehending the work. It is to be re- gretted that Chuquet did not have in De la Roche an interpreter acting with sympathy and full understanding. De la Roche mentions the Italian Luca Pacioli. De la Roche attracted little attention from writers antedating the nineteenth century; he is mentioned by the sixteenth-century French writers Buteo and Gosselin, and through Buteo by John Wallis. He employs the notation of Chuquet, intermixed in some cases, by other notations. He uses Chuquet's p and m for plus and moins, also Chuquet's radical notation $ 2 , fi 3 , # 4 , . . . . , but gives an alternative notation: H D for # 3 , HI for # 4 , HI D for # 6 . His strange uses of the geometric square are shown further by his writing D to indicate the cube of the unknown, an old procedure mentioned by Chuquet. The following quotation is from the 1538 edition of De la Roche, where, as does Chuquet, he calls the unknown and its successive powers by the names of primary numbers, secondary numbers, etc. : 104 A HISTORY OF MATHEMATICAL NOTATIONS "... vng chascun nombre est corisidere comme quantity continue que aultrement on dit nombre linear qui peult etre appelle chose ou premier: et telz nombres seront notez apposition de une unite au dessus deulx en ceste maniere 12 l ou 13 1 , etc., ou telz nombres seront signes dung tel characte apres eux comme 12. p. ou 13.P. ... cubes que Ion peut ainsi marquer 12. 3 ou 13. 8 et ainsi 12 D ou 13 D." 1 The translation is as follows: "And a number may be considered as a continuous quantity, in other words, a linear number, which may be designated a thing or as primary, and such numbers are marked by the apposition of unity above them in this manner 12 l or 13 1 , etc., or such numbers are indi- cated a^lso by a character after them, like 12. P, or 13. P. ... Cubes one u,. 40. Tart <>i iol. GO/* ot IV la Koche's Larismethique ol may mark 12. 3 or 13. 3 and also 12 D. or 13 D." (We have here 12 X = 12x, 12. 3 =12x 3 , etc.) A free translation of the text shown in Figure 40 is as follows: "Next find a number such that, multiplied by its root, the product is 10. Solution: Let the number be x. This multiplied by V x gives V / x 3 = 10. Now, as one of the sides is a radical, multiply each side by itself. You obtain z 3 = 100. Solve. There results the cube root of 100, i.e., 1^100 is the required number. Now, to prove this, multiply flOO by 1/100. But first express I^IOO as i/ , by multiplying 100 by itself, and you have 1^10,000. This multiplied by V/100 gives V 7 1,000,000, which is the square root of the cube root, or the cube root of the square root, or 1/1,000,000. Extracting the square root gives 1^1,000 which is 10, or reducing by the extraction of the cube root gives the square root of 100, which is 10, as before." 1 See an article by Terquem in the Nouvelles annales de math&matiques (Ter- quem et Gerono), Vol. VI (1847), p. 41, from which this quotation is taken. For extracts from the 1520 edition, see Boncompagni, op. tit., Vol. XIV (1881), p. 423. INDIVIDUAL WRITERS 105 The end of the solution of the problem shown in Figure 41 is in modern symbols as follows: first 1/34+7 . x x+l x+4 2-Jx [i.e., x = 1/34+4] -T: 1% i ** F . ' '" " ^ 8x+lS 4 16_ 16 1/34+4 second. iiviifv* i 'w^^^'fw|n^' : ^''^^^ p ^'TO ( jffTf i T| T ^ff i^'i K , ^v',,1 $i^^P^W r , 'v;'^^^^,^ ^tey::i^^^ 7 iite^ ~jyi4"-w"^^ ,. MM v- ,, 1 'i-**M-.ltT* l f*_*JLJiiJ I . , -, - - $F w ^w&yf^*A TkfrEWSF*'' 1 "* : ^^fc% ce, . '' .,.) !!'.??*.% i , , ,' i u ^| ^^-i 1 ^,, 1 ' r ii Hl r |i ^T l '-r- <M jj^ p _ , ,; / j / ,,_X J ^'' w/-*Lu. j.U^'/^jfc'/J^J**^ 1 FIG. 41. Part of fol. 66 of De la Roche's Larismethique of 1520 ITALIAN: PIETRO BORGI (OR BORGHI) (1484, 1488) 133. Pietro Borgi's Arithmetica was first printed in Venice in 1484; we use the edition of 1488. The book contains no algebra. It displays the scratch method of division and the use of dashes in operating with fractions ( 223, 278). We find in this early printed Arithmetica the use of curved lines in the solution of problems in alligation. Such graphic aids became frequent in the solution of the indeterminate problems of alligation, as presented in arithmetics. Pietro Borgi, on the unnumbered folio 79B, solves the following problem: Five sorts of spirits, worth per ster, respectively, 44, 48, 52, 60, 66 Midi, 106 A HISTORY OF MATHEMATICAL NOTATIONS are to be mixed so as to obtain 50 ster, worth each 56 soldi. He solves this by taking the qualities of wine in pairs, always one quality dearer and the other cheaper than the mixture, as indicated by the curves in the example. 16 4 10 4 10 8 12 Then 56-44 = 12; 66-56 = 10; write 12 above 66 and 10 above 44. Proceed similarly with the pairs 48 and 60, 52 and 66. This done, add 10, 4, 10, 8, 16. Their sum is 48, but should be 50. Hence multiply each by -J--JJ- and you obtain 10 t \ as the number of ster of wine worth 44 soldi to be put into the mixture, etc* ITALIAN: LUCA PACIOLI (1494, 1523) 134. Introduction. Luca Pacioli's Summa de arithmetica geo- metria proportioni et proportionalita (Venice, 1494) l is historically important because in the first half of the sixteenth century it served in Italy as the common introduction to mathematics and its influence extended to other European countries as well. The second edition (1523) is a posthumous publication and differs from the first edition 1 Cosmo Gordon ("Books on Accountancy, 1494-1600," Transactions of the Bibliographical Society [London], Vol. XIII, p. 148) makes the following remarks on the edition of 1494: "The Summa de arithmetica occurs in two states. In the first the body of the text is printed in Proctor's type 8, a medium-sized gothic. On sig. a 1, on which the text begins, there is the broad wood-cut border and portrait- initial L already described. In the second state of the Summa, of which the copy in the British Museum is an example, not only do the wood-cut border and initial disappear from a 1, but sigs. a-c with the two outside leaves of sigs. d and e, and the outside leaf of sig. a, are printed in Proctor's type 10**, a type not observed by him in any other book from Paganino's press. There are no changes in the text of the reprinted pages, but that they are reprinted is clear from the fact that incorrect head-lines are usually corrected, and that the type of the remaining pages in copies which contain the reprints shows signs of longer use than in copies where the text type does not vary. It may be supposed that a certain number of the sheets of the signatures in question were accidentally destroyed, and that type 8 was already in use. The sheets had, therefore, to be supplied in the nearest available type." The copy of the 1494 edition in the Library of the University of California exhibits the type 10. INDIVIDUAL WRITERS 107 only in the spelling of some of the words. References to the number of the folio apply to both editions. In the Summa the words "plus" and "minus," in Italian piu and meno, are indicated by p and m. The unknown quantity was called "thing," in the Italian cosa, and from this word were derived in Germany and England the words Coss and "cossic art," which in the sixteenth and seventeenth centuries were synonymous with "algebra." As pointed out more fully later, co. (cosa) meant our x; ce. (censo) meant our x 2 ; cu. (cubo) meant our x 3 . Pacioli used the letter 1} for radix. Censo is from the Latin census used by Leonardo of Pisa and Regiomontanus. Leonardo of Pisa used also the word res ("thing"). 135. Different uses of the symbol ft. The most common use of ft, the abbreviation for the word radix or radici y was to indicate roots. Pacioli employs for the same purpose the small letter >vl, sometimes in the running text, 1 but more frequently when he is pressed for space in exhibiting algebraic processes on the margin. 2 He writes in Part I of his Summa: (Fol. 70) 5.200. for 1/200 (Fol. 119B) ft .cuba. de .64. for (Fol. 182A)* R.relato. for fifth root (Fol. 182A) ft ft ft. cuba. for seventh root (Fol. 86A) ft .6.7n.ft.2. for 1/6-1/2 (Fol. 131A) ft ft.120. for 1/120 (Fol. 182A) ft. cuba. de ft. cuba. for sixth root (Fol. 1824) ft ft. cuba. de ft. cuba. for eighth root. The use of the fty. for the designation of the roots of expressions con- taining two or more terms is shown in the following example: (Fol. 149A) &v. ft.20Jr.m4. for 1/1/20} - The following are probably errors in the use of fty.: (Fol. 93A) fty. 50000.rn.200. for 1/50,000-200 , (Fol. 93A) R Rv. 50000.m.200. for ^50,000-200 . In combining symbols to express the higher roots, Pacioli uses the additive principle of Diophantus, while in expressing the higher powers 1 Part I (1523), fol. 86 A. */Wd., fol. 124^1. 3 On the early uses of radix relata and primo relaio see Enestrom, Bibliolhcca mathematica, Vol. XI (1910-11), p. 353. 108 A HISTORY OF MATHEMATICAL NOTATIONS he uses the multiplication principle of the Hindus. Thus Pacioli indicates the seventh root by R R R. cuba. (2+2+3), but the eighth power by ce.cexe. (2X2X2). For the fifth, seventh, and eleventh powers, which are indicated by prime numbers, the multiplication principle became inapplicable. In that case he followed the notation of wide prevalence at that time and later: p?r? (primo relato) for the fifth power, 2?r? (secundo relato) for the seventh power, 3?r? (terzo relato) for the eleventh power. 1 Whenever the additive principle was used in marking powers or roots, these special symbols became super- fluous. Curiously, Pacioli applies the additive principle in his nota- tion for roots, yet does not write R.I} cuba (2+3) for the fifth root, but I}, relata. However, the seventh root he writes R R R. cuba (2+2+3) and not 722?r?. 2 136. In other parts of Pacioli's Sumrna the sign 1} is assigned alto- gether different meanings. Apparently, his aim was to describe the various notations of his day, in order that readers might select the symbols which they happened to prefer. Referring to the prevailing diversity, he says, "tante terre: tantc vsanzo." 3 Some historians have noted only part of Pacioli's uses of R, while others have given a fuller account but have fallen into the fatal error of interpreting certain powers as being roots. Thus far no one has explained all the uses of the sign 1} in Pacioli's Summa. It was Julius Roy Pastor arid Gustav Enestrom who briefly pointed out an inaccuracy in Moritz Cantor, when he states that Pacioli indicated by R 30 the thirtieth root, when Pacioli really designated by R .30? the twenty-ninth power. This point is correctly explained by J. Tropfke. 4 We premise that Pacioli describes two notations for representing powers of an unknown, x 2 , x 3 , . . . . , and three notations for x. The one most commonly used by him and by several later Italian writers of the sixteenth century employs for x, x 2 , x 3 , x 4 , x 5 , a; 6 , x 7 , . . . . , the abbreviations co. (cosa), ce. (cemo), cu. .(cubo), ce.ce. (censo de censo), p?r? (primo relato), ce.cu. (censo de cubo), 2?r? (secundo relato), . . . , 5 Pacipli's second notation for powers involves the Use of I}, as al- ready indicated. He gives: R.p? (radix prima) for #, R.2? (radix secunda) for x, R.3? (radix terza) for x 2 , . . . . , #.30? (nono relato) for x 29 . 8 When Enestrom asserts that folio &tB deals, not with roots, 1 Part I, fol. 67. 2 Ibid., fol. 1824. 8 Ibid., fol. Q7B. 4 Op. cit. (2d ed.), Vol. II (1921), p, 109. 6 Op. cit., Part I, fol. 67. Ibid. INDIVIDUAL WRITERS 109 out exclusively with the powers x, x, # 2 , . . . . , x 29 , he is not quite accurate, for besides the foregoing symbols placed on the margin of the page, he gives on the margin also the following: "Rx. Radici; R R. Radici de Radici; Rv. Radici vniuersale. Ouer radici legata. voi dire radici vnita; R. cu. Radici cuba; $? quantita." These ex- pressions are used by Pacioli in dealing with roots as well as with powers, except that Rv. is employed with roots only; as we have seen, it signifies the root of a binomial or polynomial. In the foregoing two ases of 5, how did Pacioli distinguish between roots and powers? The Drdinal number, pn'wa, secunda, terza, etc., placed after the 5, always signifies a "power," or a dignita. If a root was intended, the number effected was written after the 5; for example, 5.200. for 1/200. In olio 143AB Pacioli dwells more fully on the use of R in the designa- tion of powers and explains the multiplication of such expressions as R. 5? via. R. ll?/a R. 15 a , i.e., x*Xx w = x u . In this notation one looks in vain for indications of the exponential concepts and recognition of :.he simple formula a m *a n = a m+n . Pacioli's results are in accordance with the formula a m a n = a m + n ~ 1 . The ordinal numbers in R 11, etc., exceed by unity the power they represent. This clumsy designation :nade it seem necessary to Pacioli to prepare a table of products, occupying one and one-half pages, and containing over two hundred md sixty entries; the tables give the various combinations of factors whose products do not exceed x 29 . While Enestrom and Rey Pastor lave pointed out that expressions like 6.28? mark powers and not *oots, they have failed to observe that Pacioli makes no use whatever 3f this curious notation in the working of problems. Apparently his aim in inserting it was encyclopedia!. 137. In working examples in the second part of the Summa, Pacioli exhibits a third use of the sign R not previously noted by historians. There R is used to indicate powers of numbers, but in a :nanner different from the notation just explained. We quote from the Summa a passage 1 in which R refers to powers as well as to roots. Wliich is meant appears from the mode of phrasing: "... 5.108. e }uesto mca con laxis ch' 5.16. fa. 5-1728 piglia el .. cioe recca .3. a. 5. a .9. parti .1728 in. 9. neuien. 192. e. 5J-92 " (.'. 1/108 and mul- tiplying this with the axis which is 1/16 gives 1/1,728. Take {-, i.e., rising 3 to the second power gives 9; dividing 1,728 by 9 gives 192, ind the 1/192. . . . .) Here "recca. 3. a. 5. fa. 9." identifies 5 with i power. In Part I, folio 186A, one reads, "quando fia recata prima. 1. * Ibid., Part II, fol. 12 B. 110 A HISTORY OF MATHEMATICAL NOTATIONS co. a. R. fa. 1. ce" ("raising the x to the second power gives a; 2 "). Such phrases are frequent as, Part II, folio 72B, "reca. 2. a. R. cu. fa. 8" ("raise 2 to the third power; it gives 8"). Observe that R. cu. means the "third" power, while 1 R. 3? and R. terza. refer to the "second" power. The expression of powers by the Diophantine additive plan (2+3) is exhibited in "reca. 3. a. R R. cuba fa. 729" ("raise 3 to the fifth power; it gives 729"). 2 A fourth use of R is to mark the unknown x. We have previously noted Pacioli's designation of x by co. (cosd) and by R. 2?. In Part II, folio 155, he gives another way: "la mita dun censo e .12. dramme: sonno equali a .5.R. E questo come a dire .10. radici sonno equali a vn censo e. 24. dramme" ("Half of x 2 and the number 12 are equal to 5x; and this amounts to saying Wx are equal to x 2 and the number 24"). In Part I, folio GO/?, the sign R appears on the margin twice in a fifth role, namely, as the abbreviation for rotto ("fraction"), but this use is isolated. From what we have stated it is evident that Pacioli employed R in five different ways; the reader was obliged to watch his step, not to get into entanglements. 138. Sign of equality. Another point not previously noted by historians is that Pacioli used the dash ( ) as a symbol for equality. In Part I, folio 91 Aj he gives on the margin algebraic expressions relat- ing to a problem that is fully explained in the body of the page. We copy the marginal notes and give the modern equivalents: Summa (Part I, fol. 91A) Modern Equivalents p? 1. co. m. 1. p? 1st x y 3? 1. co. p. 1. $? " 3d x +y 1. co. m. 1. ce. de. #5? _ 36 z 2 -?/ 2 = Rv. 1. ce. m 36 _ L ce. de ^ Valor quantitatis. the value of y . p? 1. co, m Rv. 1. ce. m 36 1st Z- T/x 2 -36 2? 6 2d 6 3? 1. co. p Rv. 1. ce. m 36 . 3d x+V / x 2 ~36 2. co. p. 6. 216 2x+6 =216 2. co. 210 2x =210 Valor rei. 105 Value of a; 105 1 Part I, fol. 67. 2 Part II, fol. 72B. INDIVIDUAL WRITERS 111 Notice that the co. in the third expression should be ce., and that the .1. ce. de $& a in the fourth expression should be .1. co. de #K Here, the short lines or dashes express equality. Against the validity of this interpretation it may be argued that Pacioli uses the dash for several different purposes. The long lines above are drawn to separate the sum or product from the parts which are added or multiplied. The short line or dash occurs merely as a separator in expressions like Simplices Quadrata 3 _ 9 in Part I, folio 39A. The dash is used in Part I, folio 54 B, to indicate multiplication, as in 14 15 where the dash between 5 and 7 expresses 5X7, one slanting line means 2X7, the other slanting line 5X3. In Part II, folio 37.A, the dash represents some line in a geometrical figure; thus d 3 fc means that the line dk in a complicated figure is 3 units long. The fact that Pacioli uses the dash for several distinct purposes does not invalidate the statement that one of those purposes was to express equality. This interpretation establishes continuity of notation between writers pre- ceding and following Pacioli. Regiomontanus, 1 in his correspondence with Giovanni Bianchini and others, sometimes used a dash for equal- ity. After Pacioli, Francesco Ghaligai, in his Pralica d'arithmetica, used the dash for the same purpose. Professor E. Bortolotti informs me that a manuscript in the Library of the University of Bologna, probaby written between 1550 and 1568, contains two parallel dashes (=) as a symbol of equality. The use of two dashes was prompted, no doubt, by the desire to remove ambiguity arising from the different interpre- tations of the single dash. Notice in Figure 42 the word cosa for the unknown number, and its abbreviation, co.; censo for the square of the unknown, and its con- traction, ce.; cubo for the cube of the unknown; also .p. for "plus" and .ra. for "minus." The explanation given here of the use of cosa, censo, cubo, is not without interest. 1 See Maximilian Curtze, Urkunden zur Geschichte der Malhematik im Mittel- alter und der Renaissance (Leipzig, 1902), p. 278. 112 A HISTORY OF MATHEMATICAL NOTATIONS The first part of the extract shown In Figure 43 gives J/1/40+6+ ^1/406 and the squaring of it. The second part gives ^1/20+2 +1/1/202 and the squaring of it; the simplified result is given as 1/80+4, but it should be 1/80+8. Remarkable in this second example is the omission of the v to express vniversale. From the computation as well as from the explanation of the text it appears that the first B was intended to express universal root, i.e., */ 1/20+2 and not 1/20+2. FIG. 42. Part of a page in Luca Pacioli's Summa, Part I (1523), fol. ITALIAN: F. GHALIGAI (1521, 1548, 1552) 139. Ghaligai's Pratica d'arithmetica 1 appeared in earlier editions, which we have not seen, in 1521 and 1548. The three editions do not differ from one another according to Riccardi's Biblioteca matematica italiana (I, 500-502). Ghaligai writes (fol. =cubo=| | | , x r =pronico = ll = tronico- r-j, x n dromico= -- . He uses the m for "minus' : and the $ and 6 for "plus," but frequently writes in full piu and meno. 1 Pratica d'ariihmetica di Francesco Ghaligai Florentine (Nuouamente Riuista, & con somma Diligenza Ristampata. In Firenze. M.D.LlI). INDIVIDUAL WRITERS 113 Equality is expressed by dashes ( ) ; a single dash ( ) is used also to separate factors. The repetition of a symbol, simply to fill up an interval, is found much later also in connection with the sign of equality ( = ). Thus, John Wallis, in his Mathesis universal. ([Oxford, 1657], p. 104) writes: 1+2-3= - -0. mea potS rationale t mediV el:t?5/iftfton, iRadijc qnti bino* ifnea potes rprwle z irronafe* iRadijc qm'ntt binomij* 2 O 2 kflgjk 2 O. FIG. 43. Printed on the margin of fol. 123/i of Pacioli's Summa, Part I (1523). The same occurs in the edition of 1494. Ghaligai does not claim these symbols as his invention, but ascribes them to his teacher, Giovanni del Sodo, in the statement (folio 71JB): "Dimostratione di 8 figure, le quale Giovanni del Sodo pratica la sua Arciba & perche in parte terro 'el suo stile le dimos- treto.' " l The page shown (Fig. 45) contains the closing part of the I 0p. tit. (1552), fols. 2B, 65; Encstrom, Bibliolhcca mathcmatica, 3. S., Vol. VIII, 1907-8, p. 90. 114 A HISTORY OF MATHEMATICAL NOTATIONS solution of the problem to find three numbers, P, S, T, in continued proportion, such that S* = P+T, and, each number being multiplied !' j , , r*- v r ^ Tft^rj* , * 'i if^'V^', ifrl^vt/ste^ 1 *^ ' 4' ,vVj4 irlfTC*^'. , ,, ?, j, --^feKfi^ ; l : ;?l'^|f^^ ;.'; ; fMriotuleitit ** rUt*"*)"** H*HfW^lf Y*~ TPfpj H^ ""'^jr, ,T= -T "' -~ rT' T" - f ' ' | ^| ,J | 1^,1* | [ 1 - r * ' ' ' "" '' -l " ! " J '" a 'ia'^^tife^i^ttffl^i 1 ^ ft&ft^tt^ ^.;)/-;. ^ '",; ; ' ; :;>cQ^B!* l 4tew^f^ dttl^oMHatotiaaa ;^^ t lts-IMr t^l, j^ 1 '- tfci * 'rt jp*i|' ;l | FIG. 44. Part of fol. 72 of Ghaligai's Praticad'arithmetica (1552). This exhib- its more fully his designation of powers. INDIVIDUAL WRITERS 115 by the sum of the other two, the sum of these products is equal to twice the second number multiplied by the sum of the other two, plus 72. Ghaligai lets S = 3co or 3x. He has found x = 2, and the root of x 2 equal to 1/4 . The translation of the text in Figure 45 is as follows: "equal to 1/4, and the 'I/a; 4 is equal to 1/16, hence the first quantity was 18-1/288, and the second was 6, and the third 18+1/288. S. 3x P. and T. 9x 2 18o; 2 +6x,X3x . P. T. P. 4 ? 2 x 2 \ 4 --- / Value of x which is 2 18 1/324 P. was 18-1/288 36 S. was 6 1/288 T. was 18+1/288 Proof 24+1/288 24-1/288 18+1/288 432+1/93,312-288 432+1/165,888-288 288 - 1/165,888 288 - l/'9pT2 144 144 144+1/93,312^ 18-1/288 -1/165J888. 18+1/288 + 1/165,* 144-1/93,312 =36,X6 Gives 288 216,X2 216 432 Gives 504 72 As it should 504." 116 A HISTORY OF MATHEMATICAL NOTATIONS ffl^ ' 'i " L i *"*>-** **** *#**W , ',' '' '' ' "^ ' ' 7n* 1 pdi4 9Sil^lCitf*w^W'W^^-^>,T,~,~ J" ^fei..^''^'.''^" '';;' 'i^" l ,i i ," i ''C'; 1 ;*"'-:- ; l '4.-",'- w - l - *". f?;,SjE?r ?v!;t:^^^^^f,n??:a - '" FIG. 45, Ghaligai's Praticad'arithmtica (1552), fol. 108 INDIVIDUAL WRITERS 117 The following equations are taken from the same edition of 1552: Translation (Folio 110) i D di D m \ di D 1 D {x 4 -\x 2 = x* i D di D 1.1 di D \x*=l\x* in -ii-fl iz 2 =u (Folio 113) iD D w 4 D 4 D }a; 4 -4a; 2 = 4z 2 i a a 8 a }z 4 =8z 2 Ghaligai uses his combinations of little squares to mark the orders of roots. Thus, folio 84#, # D di 3600 die e 60, i.e., 1/3,600 = 60; folio 727?, laH\m di 8 ditiamo 2, i.e., f/8 = 2; folio 737?, 7 | j di 7776 for f 7,776; folio 73B, 7 rT I di D di 262144 for \/ 262,144. ITALIAN: HIERONYMO CARDAN (1539, 1545, 1570) 140. Cardan uses p and m for "plus" and "minus" and $ for "root." In his Pradica arithmeticae generalis (Milano, 1539) he uses Pacioli's symbols nu. y co., ce., cu., and denotes the successive higher powers, cc.ce., Rel. p., cu.ce., Rel. 2., ce.ce.ce., cu.cu., ce. Rel. 1 How- ever, in his Ars magnet (1545) Cardan does not use co. for x, ce. for # 2 , etc., but speaks of "rem ignotam, quain vocamus positionem," 2 and writes 60+20z=100 thus: "60. p. 20. positionibus acqualia 100." Farther on 3 he writes x 2 +2o; = 48 in the form "1. quad. p. 2. pos. aeq. 48.," x 4 in the form 4 "1. quadr. quad.," z 5 +6z 3 = 80 in the form 5 "r. p m p. 6. cub. 80," z 5 = 7z 2 +4 in the form 6 "r m p m 7. quad. p. 4." Observe that in the last two equations there is a blank space where we write the sign of equality ( = ). These equations ap- pear in the text in separate lines; in the explanatory text is given aequale or aequatur. For the representation of a second unknown he follows Pacioli in using the word quantitas, which he abbreviates to quan. or qua. Thus 7 he writes 7x+37/ = 122 in the form "7. pos. p. 3. qua. aequal. 122." Attention should be called to the fact that in place of the p and m, given in Cardan's Opera, Volume IV (printed in 1663), one finds in Cardan's original publication of the Ars rnagna (1545) the signs p: 1 Ilieronymi Cardani operum tomvs quartvs (Lvgdvni, 1663), p. 14. 2 Ibid., p. 227. *Ibid., p. 231. *Ibid. 4 Ibid., p. 237. 6 Ibid., p. 239. 7 Ars rnagna in Operum tomvs quartvs, p. 241, 242. 118 A HISTORY OF MATHEMATICAL NOTATIONS and m:. For example, in 1545 one finds (5+ 1/ 15) (5 I/ 15) = 25 (15) =40 printed in this form: " 5p: R m: 15 5m: & m: 15 25w:m: 15 qd est 40 ," while in 1663 the same passage appears in the form: " 5. p. I}, m. 15. 5. m. I}, m. 15. 25. m. m. 15. quad, est 40. ",* 141. Cardan uses 1} to mark square root. He/ employs 2 Pacioli's radix vniversalis to binomials and polynomials, thus "R.V.7. p R. 4. vel sic (R) I3.p R. 9." for 1/7+ 1/4 or 1/13+V 9; "#.7.10.p.#.16.p.3.p 8.64." for V 10 +1/16+3 +1/64. Cardan proceeds to new nota- tions. He introduces the radix ligata to express the roots of each of the terms of a binomial; he writes: "LR. 7. pR. 10." 3 for 1/7+ 1/10. This L would seem superfluous, but was introduced to dis- tinguish between the foregoing form and the radix distincta, as in "R.D. 9 p. R. 4.," which signified 3 and 2 taken separately. Accord- ingly, "#.D. 4. p. 8. 9.," multiplied into itself, gives 4+9 or 13, while the "fi.L. 4. p. #. 9.," multiplied into it/self, gives 13+1/144 = 25. In later passages Cardan seldom uses the radix ligata and radix dis- tincta. In squaring binomials involving radicals, like "R.V.L. R. 5. p. R. 1. m R.V.L. R. 5. m R. 1.," he sometimes writes the binomial a second time, beneath the first, with the capital letter X between the two binomials, to indicate .cross-multiplication. 4 Of interest is the follow- ing passage in the Regula aliza which Cardan brought out in 1570: "Rp: est p: R m: quadrata nulla est iuxta usum communem" ("The square root of a positive number is positive ; the square root of a nega- tive number is not proper, according to the common acceptation"). 6 1 Sec Tropfkc, op. tit., Vol. Ill (1922), p. 134, 135. 2 Cardan, op. til., p. 14, 16, of the Practica arithmeticae of 1539. 3 Ibid., p. 16. 4 Ibid., p. 194. 5 Op. tit. (Basel, 1570), p. 15. Reference taken from Enestrom, Bibliotheca mathematica, Vol. XIII (1912-13), p. 163. INDIVIDUAL WRITERS 119 However, in the Ars magna 1 Cardan solves the problem, to divide 10 into two parts, whose product is 40, and writes (as shown above) : " 5. p ft. m. 15. 5. m R. m.. 15. 25 ra.w. 15. quad, cst 40 . "tCjill'fft i'lfJPiyi* " TT^M r T 1 !^ '"j^""' Y'I"""'"' ' | -f * \m, t WWtfi..* " .ffiKL W'l&J'V* ik!l:. ..'4cnfc4 , . ' ' . FIG. 46. ^Part of a page (255) from the Ars magna as reprinted in H. Cardan's Operum tomvs quartvs (Lvgdvni, 1663). The Ars magna was first published in 1545. 1 Operum tomvs quartvs, p. 287. 120 A HISTORY OF MATHEMATICAL NOTATIONS In one place Cardan not only designates known numbers by letters, but actually operates with them. He lets a and b stand for any given numbers and then remarks that R , is the same as , -,- , xi Va . is the same as = . Vb Figure 46 deals with the cubic x 3 +3x 2 = 21. As a check, the value of x, expressed in radicals, is substituted in the given equation. There are two misprints. The 226 \ should be 256 \. Second, the two lines which we have marked with a stroke on the left should be omitted, except the m at the end. The process of substitution is un- necessarily complicated. For compactness of notation, Cardan's symbols rather surpass the modern symbols, as will be seen by com- paring his passage with the following translation: "The proof is as in the example o; 3 +3:c 2 = 21. According to these rules, the result is Sl + V+^-V.-l. The cube [i.e., # 3 ] is made up of seven parts: 12- ^4,846j + 1/23,487,833}:- ^4,846 \ \- 1/23,487,833 \ + ^46,041 1+ 1/271197776^950? - V 2^096,286, 117 ^ + ^46,041 2 + 1/2,096,354 ~1 80 jl - ^2,1 19,776,950J + ^256^+1/65^0631 + ^256^- 1/65,063] . "The three squares [i.e., 3x 2 ] are composed of seven parts in this manner: 9+ ^4,8461+1/23^87,8331 | - V 23,487,833J - 256^+ V 65^)63 1 -^256 ^1/65^0631 - 1^256 ? 2 - ^256 \- 1/657063}. Now, adding the three squares with the six parts in the cube, which are equal to the general cube root, there results 21, for the required aggregate." 1 Deregulaaliza (1570), p. 111. Quoted by Enestrom, op. cit. t Vol. VII (1906-7), p. 387. INDIVIDUAL WRITERS 121 In translation, Figure 47 is as follows: "The Quaestio VIII. "Divide 6 into three parts, in continued proportion, of which the sum of the squares of the first and second is 4. We let the first be the Q_y A s T i o VIII. Fac ex 6. tres paries, in conrinua proper- tionc, cjuarum quadrata primx & iecundae iun<ta jfimiil facianc 4. ponemus pnmain i. poiitionenV, quadratum eius dt i. qua- dratum , txiiduum igicur ad 4. ell quadra- tum fccundx quantitatis, id til 4. m. i qua- dratb , huius radicem, & i poiitionem dc- trahe ex 6 . habebis cectiatn quantitacem> vc vidcsj ^juaicduCla prima in tcdam> ha- i .pof. I v. $.4. m. r. quad. \6. m. i. pof. m. ^. v. 4, m. i . quad. 6. pof. m. i. quad. m. ty. v. 4. quad. ni. i. quad/quad. 4. i 6. pof.m. $t. v. 4. quad* m. i. quad. quad. 6. pof. m. 4. xquaj. $t. v. 4. quad. m. i . quad. quad. 56. quad p. 1 6-. m. 48. pof. xquantur 4. quad. m. i. quad. quad. -j i . quad, quad.p. 3 1. quad. p. i $6- lia 48. poCp 140* \ i. quad. p. 16. p. i. quad. quad, xqua- lia 48. pof. , FIG. 47. Part of p. 297, from the Ars magrui, as reprinted in II. Cardan's Operum tomvs quartvs (Lvgdvni, 1663). 1. position [i.e., x]; its square is 1. square [i.e., x 2 ]. Hence 4 minus this is the square of the second quantity, i.e., 41. square [i.e., 4 # 2 ]. Subtract from 6 the square root of this and also 1. position, and you will have the third quantity [i.e., 6 x 1/4 x 2 ], as you see, because the first multiplied by the third . . . . : 122 A HISTORY OF MATHEMATICAL NOTATIONS 4 ~x 2 | G-x-l/4-x 2 36x 2 +16-48x = 4x 2 - 32x 2 +16+x 4 = Ix 4 +32x 2 +256 = 48x+240 ITALIAN: NICOLO TARTAGLIA (1537, 1543, 1546, 1556-60) 142. Nicolo Tartaglia's first publication, of 1537, contains little algebraic symbolism. He writes: "Radice .200. censi piu .10. cose" for V / 2(X)x 2 + lOx, and "trouamo la cosa ualer Radice .200. men. 10." for "We find x== 1/200- 10." 1 In his edition of Euclid's Elements 2 he writes "ft ft ft ft" for the sixteenth root. In his Qvesiti 3 of 1546 one reads, "Sia .1. cubo de censo piu .48. equal a 14. cubi" for "Let z+48 = 14x 3 ," and "la ft. cuba de .8. ualera la cosa, cioe. 2." for "The 1^8 equals x, which is 2." More symbolism appeared ten years later. Then he used the p and m of Pacioli to express "plus" and "minus," also the co., ce., cu., etc., for the powers of numbers. Sometimes his abbreviations are less intense than those of Pacioli, as when he writes 4 men instead of m, or 5 ccn instead of ce. Tartaglia uses ft for radix or "root." Thus "la ft # di A 6 i" 6 "la ft cu. di J e |," 7 "la ft rel. di ^ e V 8 "la ft cen. cu. di ,'4 6 i," 9 "la ft cu. cu. di 6 b 6 i" 10 "la ft terza rel. di 2 ^ e |." n 143. Tartaglia writes proportion by separating the three terms which he writes down by two slanting lines. Thus, 12 he writes "9// 5//100," which means in modern notation 9:5 = 100:x. For his occasional use of parentheses, see 351. 1 Nova scientia (Venice, 1537), last two pages of "Libro secondo." 2 Evclide Megareme (Venice, 1569), fol. 229 (1st ed., 1543). 3 Qvesiti, ct invent iojii (Venice, 1540), fol. 132. 4 Scconda parle del general trattalo di nvm.eri, et misvri de Nicolo Tartaglia (Venice, 1556), fol. 88. 6 Ibid., fol. 73. 9 Ibid., fol. 47#. c Ibid., fol. 38. 10 76?:^., fol. 60. 7 Ibid., fol. 34. Ibid., fol. 68. 8 Ibid., fol. 43. 12 Ibid., fol. 162. INDIVIDUAL WRITERS 123 On the margin of the page shown in Figure 48 are given the sym- bols of powers of the unknown number, viz., co., ce., etc., up to the twenty-ninth power. In the illustrations of multiplication, the absolute number 5 is marked "5w/0"> the after the solidus indi- cates the dignitd or power 0, as shown in the marginal table. His i' r 11 i.^J!L.,J! JL ft^^ .1 tt.-. JjiiL..- _ JL_ joUukJ., i,t.J,Li T Ti ' , 14-1"- TI ^mn JrKKiK^^ ^ ppiv^wi < rnrriT^ - ', SssttB^ms'Kurtb'^ jfff^i, f r 4;*?^^Vi., MI Jr ,*H~'i *f^ 1 *W7"*^ l 7ra 1 ^^ 1 WrW , Jl fj i "''" v: * ; ^Sjfjs^^''--' "^ , ,/; r i lr : p ,.," h 'i' >>n i : ' , ; ,; i [1; " P if ^^'' ''r ^ UK * ^St ' J I J ' '/ K i'ii *' ' ' SUtfj- 1 * i~L uh'i* * ^"ih Ji **>" ~f 1 = IA 1 ' lAijS^ifcjBiw J r i*Jfc*J'*-'>fc* fc ''* R{ * 1L * FIG. 48. Part of a pa^e from Tarlaglia/s La .sr.s-/a ^arte dd general trattato de nvmeri, et misvre (Venice, 1560), fol. 2. illustrations stress the rule that in multiplication of one dignitd by another, the numbers expressing the dignitd of the factors must be added. ITALIAN: RAPAELE BOMBELLI (1572, 1579) 144. Bombelli's Ualgebra appeared at Venice in 1572 and again at Bologna in 1579. He used p. and m. for "plus" and "minus." 124 A HISTORY OF MATHEMATICAL NOTATIONS Following Cardan, Bombelli used almost always radix legata for a root affecting only one term. To write two or more terms into one, Bombelli wrote an L right after the # and an inverted J at the end of the expression to be radicated. Thus he wrote: & L 7 p. ft 14 J for our modern 1/7+1/14, also^Rg L Re L RqQ8 p.2JmRcLRq68m 2 JJ for the modern l/{^(l/68+2) - ^(1/68-2) }. ' I,;, ' "^ l ' lL ' h: '" !*f^ ' jJ(Bidit.^k'i*,iM*iiitl>tttWi*i*i,li ll ' ' 7 1 ', 1 ' '^ '''' ' ' - ' ;'' ; '^ "i 1 J ! ' |^ ; " i|F| , h ;j' ^ , ^,^^1^^^^ , Jr , ^ ,^ ^ r i'"^ ' " : '" '''"" ' i ' ' * ^ ; ,i-. a f > ; ,;;' , ' y 1 , - ', it c* t * * *_ - ' ,. wfett IfMfW t 9 Ha*- ir '"' h r L-^^iSE^2to^^^ r,;?;,;,;' ' : ,' i- 1 '' ' t ( _ 1 ' ,,;'. v ' ' [ r _ ~ f FIG. 49. Part of a page from Tartaglia's La sesta partc del general trattalo de nvmeri, et misvre (Venice, 1560), fol. 4. Shows multiplication of binomials. Ob- serve the fancy .p. for "plus." For "minus" he writes here me or men. An important change in notation was made for the expression of powers which was new in Italian algebras. The change is along the line of what is found in Chuquet's manuscript of 1484. It is nothing less than the introduction of positive integral exponents, but without writing the base to which they belonged. As long as the exponents were applied only to the unknown x, there seemed no need of writing the x. The notation is shown in Figure 50. * Copied by Cantor, op. dt., Vol. II (2d ed., 1913), p. 624, from Bombelli's L'algebra, p. 99. INDIVIDUAL WRITERS 125 : 4: -":-\,:--i ->x- N. 'm^ , ' refti > e , " : ' 11 4 |K 4 4 & * ( : v .-V--H;'- :: "-.;^ -"/ ; " 4. V - '. t * -_. . - Jl '"'" "'-"*"* " - : ' - i* -"-'-" !. ** _ T \- : ii_ ; -:" - " "-- : ": -^V ; V ' - m "-10*- , '" , -.-"". t - -& -'' - ' ' ^*-' :: ' - "j? - -- W - r "- '";, - " . ( <^ '--" % * -r - " ' - ' p id. ""-"V " .' '- ' .. ^ - ;;."- 'ft- - f = : . - = -. - ,: ;.' . . ; ^r t .- ;^_ '-- .- . ' ". ;,/ '- ; . . ' " - - " ..' '.-/-;-_ . -"- :-';. ' -.. --'_-: ;;-.-f :;;-'*- ?v.\ '-,-'',-'- ~ 9. - "- " -^" ; : ' ^: / -'- ; -' - " " : '-' ' :.--^--^^^^:: l ^;^:A-^^ : .. y ;- ^: l .-'-;'^0. :; .-"^:: : -" .;;-. . -..-. ,. : - T . : ; -;',,---', W'V"-y ; _^:^- > '/'-. rr^x-,,^,.::^. 1 "-"- 1 ". " ' ."-,-- : ',- -". ;_, -., . ---.:.- - Li --^-i j - - ', : - ;-..-'-"-.-> -." "-.--.,-. - te> -- >"*-- '' : -- '-"--" ' - &^"- : :'%S-^ : ^ "'-" - ; '- : ""- ' -/- -..*' - ^ - . .-: - - -- I* -." \ - : -"-";-; -/-'."."-"."- : FIG. 50.- From Bumbelli's L r algebra (1572) 126 A HISTORY OF MATHEMATICAL NOTATIONS In Figure 50 the equations are: 4x2+40;, Bombelli expressed square root by R. q. y cube root by R. c., fourth root by R R. q., fifth root (Radice prima incomposta, ouer relata) by R. p. r., sixth root by R. q. c., seventh root by R. s. r., the square root of a polynomial (Radice quadrata legato) by R. q. L J ; the cube root of a polynomial (Radice cubica legato) by R. c. L J. Some of these symbols are shown in Figure 51. He finds the sum of ^72 V 7 1,088 and ^ V 4;352+16 to be <* 232+] ^312. The first part of the sentence preceding page 161 of Bombelli's Algebra, as shown in Figure 51, is "Sommisi R. c. L R. q. 4352 .p. 16.J con R. c. L 72. m. R. q. 1088.J." 145. Bombelli's Algebra existed in manuscript about twenty years before it was published. The part of a page reproduced in Figure 52 is of interest as showing that the mode of expressing aggregation of terms is different from the mode in the printed texts. We have here the expression of the radicals representing x for the cubic a: 3 = 32x+24. Note the use of horizontal lines with cross-bars at the ends; the lines are placed below the terms to be united, as was the case in Chuquet. Observe also that here a negative number is not allowed to stand alone: 1069 is written 01069. The cube root is designated by R*, as in Chuquct. A manuscript, kept in the Library of the University of Bologna, contains data regarding the sign of equality ( = ). These data have been communicated to me by Professor E. Bortolotti and tend to show that ( = ) as a sign of equality was developed at Bologna inde- pendently of Robert Recorde and perhaps earlier. The problem treated in Figure 53 is to divide 900 into two parts, one of which is the cube root of the other. The smaller part is desig- I $ j; ,P? T: ] : " I ,[ r ": ;; .'fV;,.;' 1 ,;. >>'..;,;,, :;^i ^ l ;; l ^ L i^Ai|'fr .'!^ FIG. 51. Bombelli's Algebra, p. 161 of the 1579 impression, exhibiting the calculus of radicals. In the third line of the computation, instead of 18,415,616 there should be 27,852,800. Notice the broken fractional lines, indicating difficulty in printing fractions with large numerators and denominators. 128 A HISTORY OF MATHEMATICAL NOTATIONS nated by a symbol consisting of c and a flourish (probably intended for co) . Then follows the equation 900 fn Ico = leu, (our 900 - x = z 3 ) . One sees here a mixture of two notations for x and x 3 : the notation co and cu made familiar by Luca Pacioli, and Bombelli's exponential notation, with the 1 and 3, placed above the line, each exponent resting in a cup. It is possible that the part of the algebra here photo- graphed may go back as far as about 1550. The cross-writing in the photograph begins: "in libro vecchio a carte 82: quella di far di 10 due parti: dice messer Nicolo che Pona e & 43 p 5 m RIS: et 1'altra il resto sino a 10, cioe 5 m R 43 p. & 18." This Nicolo is supposed to be Nicolo Tartaglia who died in 1557. The phrasing "Messer Nicolo" implies, so Bortolotti argues, that Nicolo was a living contemporary. If these contentions are valid, then the manuscript in question was written in 1557 or earlier. 1 I Jt Fi(]. /)2. From the manuscript of the Algebra of Bombelli in the Comunale Library of Bologna. (Courtesy of Professor E. Bortolotti, of Bologna.) The novel notations of Bombelli and of Ghaligai before him did not find imitators in Italy. Thus, in 1581 there appeared at Brescia the arithmetic and mensuration of Antonio Maria Visconti, 2 which follows the common notation of Pacioli, Cardan, and Tartaglia in designating powers of the unknown. GERMAN: IOHANN WIDMAN (1489, 1526) 146. Widman's Behennde vnnd hubsche Rechnug auff alien Kauff- manschafften is the earliest printed arithmetic which contains the signs plus (+) and minus ( ) (see 201, 202). 1 Since the foregoing was written, E. Bortolotti has published an article, on mathematics at Bologna in the sixteenth century, in the Periodico di Matcmaliche (4th ser., Vol. V, 1925), p. 147-84, which contains much detailed information, and fifteen facsimile reproductions of manuscripts exhibiting the notations then in use at Bologna, particularly the use of a dash ( ) and the sign ( = ) to express equality. 2 Antonii Mariae Vicecomitis Civis Placentini practica numerorum & mensu- rarum (Brixiac, 1581). INDIVIDUAL WRITERS 129 ,kw*j '*>'- r ;&;/#iV^ i * ' '' - r tc " t *^v * 'rl j^v^lfP'^ ?.y>%^,^^p^^^R|S| -r-^-fe 'W^rl" :?i?w$i$F : : / .! "-"-."trS;',;:!,^ L - .' - "'' >' Wv*^r^ ,-"-'/_: *$*i*^i&>*- ' ,V ^ -\ !%mt-,^>|' *- 1* iw 1 . ; . - . . , ,^*1,".^| ^4^f r ' ' li N^MA4-f'. : c i .'''V^'TIyi '* s- . i; FKI. 53. From a pamphlet (marked No. 59oX, in the Library of the Uni- versity of Bologna) containing studies and notes which Professor Bortolotti con- siders taken from the lessons of Pompeo Bolognetti ([Bologna?]-1568). 130 A HISTORY OF MATHEMATICAL NOTATIONS 4 -f T 4 - 1 3 4* Jo d>eit/6tfiiimetf 4 -- 1 p fcie3Mttnrnt> 3 4- 44 IfcWWfr W<4ttff 3 4- a* ift/*<i&ifi mi* 5 -- 1 1 H> 3 -f- 44 3 -h 19 3 -*-i* -f S> 4^39^ AinUgel 14 fc. Vnb ^dijl 1 3 ma ^nt>tnad)t j i z tt.ntmrp:td> i oo H>t)<t6tjf ctftjentttcr 21 Fia. 54. From the 1526 edition of Widman's arithmetic. (Taken from D. E. Smith, Rara arithmetica, p. 40.) 7<$ 15 r> 't 1^0 f W-f f tct ynt 4S tS 9 FIG. 55. From the arithmetic of Grammateus (1518) INDIVIDUAL WRITERS 131 AUSTRIAN: HEINRICH SCHREIBER (GRAMMATEUS) (1518, 1535) 147. Grammateus published an arithmetic and algebra, entitled Ayn new Kunstlich Buech (Vienna), printed at Niirnberg (1518), of which the second edition appeared in 1535. Grammateus used the fein }ti abbiren bieqttfttttttet fined ti# 6tft.mitft:pzimrottpn>na/Teciinba mit fecuba/teitia mir tertia jc.Tnfc maobrau* <f>et (btyetttgen ale -)-if? tticfc:/t>nb / fc*r/in weUfcer fern 50 mere? en t>:ei Kf rtel CDanncm quantitet fiat an beybcn oiten-f- ebcr fofol mann ale 9 p:i.--7U 8 pti. i 14 tl. 3ft in bcr Sbern cjuntitet ~j- \jnb fn bet t)ti r rn / vnb |- tibertriff r-/ fo fol Me vnber quntitec ron ber 5>bern fubtrafrtrt tceirben/ori $ubemub:iaenfe^ f-@oaberbie wtber qua titer if? 0r5ffcr/fo fubtraf)ir bie BIdncrn to bcr 4p?.-|-iN. tofW. 4N. 60 inberob0cfaRtenqantiter wiirtfunbe rnb in t'cr vnbern -|-/pnb ubertnjf t -{-/ fo Inbrrabir cn;o x^on bcm nbcrn/nb'5iim u> b0cn fdj:cib 3(?e<54bcr/baebie rnbcrqua ruetiibei'trijft bie >bcrn/fo$tef)c etnd von bcm Anbcrn/ pub ju bum cr Hen fcige J- aid FIG. 56. From the arithmetic of Grammateus (1535). (Taken from D. E. Smith, Rara arithmetica, p. 125.) 132 A HISTORY OF MATHEMATICAL NOTATIONS plus and minus signs in a technical sense for addition and subtraction. Figure 55 shows his mode of writing proportion: 7Glb. : 13fl. = I2lb.:x. He finds x = 2fl. s. 12] |#. [l/Z.=8s., Is. = 300]. The unknown quantity x and its powers x 2 , # 3 , . . . . , were called, respectively, pri (primci), 2a. or se. (secondd), 3a. or ter. (terzd), 4a. erjfci* roitfce fcfebm/vtifeber quoci vnb &cr quo c tent P p;l aii iDatimct) inultt pUct f t>ae I?mP tayl P trt ftd)/ rii btcem pm :4 34. 44, 7 49- FKJ. 57. From the arithmetic of Grammateus (1518) or quart, (quarto), 5a. or quit, (quinta), 6a. or sex. (sexto)] N. stands for absolute number. Fig. 56 shows addition of binomials. Figure 57 amounts to the solution of a quadratic equation. In translation: "The sixth rule: When in a proportioned number [i.e., in 1, , x 2 ] three quantities are taken so that the first two added together arc equal to the third [i.e., INDIVIDUAL WRITERS 133 d+ex=fx*], then the first shall be divided by [the coefficient of] the third and the quotient designated a. In the same way, divide the [coefficient of] the second by the [coefficient of] the third and the quotient designated 6. Then multiply the half of 6 into itself and to the square add a; find the square root of the sum and add that to half of 6. Thus is found the N. of 1 pri. [i.e., the value of x]. Place the number successively in the seven-fold proportion N: x x 2 x 3 z 4 x 5 1. 7 49 343. 2,401. 16,807. Now I equate 12x+24 with 2}!Jz 2 . Proceed thus: Divide 24 by 2-J-jjx 2 ; there is obtained 10a. Divide also 12x by 2Jgx 2 ; thus arises 5jJ6. Multiplying the half of b by itself gives VsVi to which adding a, i.e., 10JJ, will yield %| 8 , the square root of which is ]; add this to half of the part 6 or -f jj, and there results the number 7 as the number 1 pri. [i.e., x]." The following example is quoted from Grammatcus by Treutlein: 1 " Gpri. + SN. Modern Symbols Durch 6z +8 5 pri. 7 N. 5z 7 30 se.+40 pri. 30z 2 +40x - 42 pri.- 56 N. -42z-56 30 se. - 2 pri. -56N." 30x 2 - 2x - 56 . In the notation of Grammateus, 9 /er.+30 se. 6 pri. +48N. stands for 9z 3 +30z 2 -6z+48. 2 We see in Grammateus an attempt to discard the old cossic sym- bols for the powers of the unknown quantity and to substitute in their place a more suitable symbolism. The words prima, seconda, etc., remind one of the nomenclature in Chuquet. His notation was adopted by Gielis van der Hoecke. GERMAN : CHRISTOFF RUDOLFF (1525) 148. RudohTs Behend vnnd Hubsch Rechnung durch die kunst- reichen regeln Algebre so gemeincklich die Coss genent werden (Strass- 1 P. Treutlein, Abhandlungen zur Geschichle der Mathematik, Vol. II (Leipzig, 1879), p. 39. 2 For further information on Grammateus, see C. I. Gerhardt, "Zur Ge- schichto der Algebra in Deutschland," Monatsbericht d. k. Akademie der Wissen- schaften zu Berlin (1867), p. 51. 134 A HISTORY OF MATHEMATICAL NOTATIONS burg, 1525) is based on algebras that existed in manuscript ( 203). Figure 58 exhibits the symbols for indicating powers up to the ninth. The symbol for cubus is simply the letter c with a final loop resembling the letter e, but is not intended as such. What appears below the symbols reads in translation: "Dragma or numerus is taken here as 1. It is no number, but assigns other numbers their kind. Radix is the fro. J)4&m audj je erne t>on f tir? twgm mit ctncm r:0f nomen t>pn anfangfc* nwf $ pftr na ct *v jtnftqat* furfpfifuitn Fio. 58. From Rudolff's Coss (1525) side or root of a square. Zensus, the third in order, is always a square; it arises from the multiplication of the radix into itself. Thus, when radix means 2, then 4 is the zensus." Adam llicse assures us that these symbols were in general use ("zcichen ader benennung Di in gemeinen brauch teglich gehandelt werdcnn"). 1 They were adopted by Adam 1 Riese's Coss was found, in manuscript, in the year 1855, in the Kirchen- und Schulbibliothek of Marienberg, Saxony ; it was printed in 1892 in the following publication: Adam Riese, sein Lcben, seine Rechenbtictor und seine Art zurechnen. Die Coss von Adam Riese, by Realgymnasialrektor Bruno Berlet, in Annaberg i. E., 1892. INDIVIDUAL WRITERS 135 Riese, Apian, Menher, and others. The addition of radicals is shown in Figure 59. Cube root is introduced in Rudolff s Coss of 1525 as follows: "Wiirt radix cubica in diesem algorithmo bedeut durch solchen character wv/> a l 8 /vw/ 8 is zu versteen radix cubica aufs 8." ("In this algorithm the cubic root is expressed by this char- acter AVS/> as /WA/8 is to be understood to mean the cubic root of 8.") The fourth root Rudolff indicated by /w/ ; the reader naturally wonders why two strokes should signify fourth root when three strokes indi- cate cube root. It is not at once evident that the sign for the fourth tyenvpi t>on communicant m jte fa: fa: /s'|jp fa: ^9? fa: V 4 <>H- - jtmpf *on wacuwafrt item ^ 4 tfti / 1 j factf S fr CP(( i^-*- ^108 FIG. 59. From Rudolff's Coss (1525) root represented two successive square-root signs, thus, \/i/. This crudeness in notation was removed by Michael Stifel, as we shall see later. The following example illustrates Rudolff's subtraction of frac- tions: 1 "1 #-2 12 148-lj ~~ von Kest 12 On page 141 of his Coss, Rudolff indicates aggregation by a dot; 2 i.e., the dot in "j/. 12+1/140" indicates that the expression is v/ 12+l/140, and not 1/12+1/140. In Stifel sometimes a second dot appears at the end of the expression ( 348). Similar use of the dot we shall find in Ludolph van Ceulen, P. A. Cataldi, and, in form of the colon (:), in William Oughtred. When dealing with two unknown quantities, Rudolff represented 1 Treutlein, "Die deutsche Coss," op. cit. t Vol. II, p. 40. *G. Wertheim, Abhandlungen zur Geschichte der Mathematik, Vol. VIII (Leipzig, 1898), p. 153. 136 A HISTORY OF MATHEMATICAL NOTATIONS the second one by the small letter q, an abbreviation for quantita, which Pacioli had used for the second unknown. 1 Interesting at this early period is the following use of the letters a, c, and d to represent ordinary numbers (folio Giij a ) : "Nim \- solchs collects | setz es auff ein ort | dz werd von lere wegen c genennt. Dar- nach subtrahier_das c vom a | das iibrig werd gesprochen d. Nun sag ich dz Vc+Vd ist quadrata radix des ersten binomij." ("Take | this sum, assume for it a position, which, being empty, is called c. Then subtract c from a, what remains call d. Now I say that Vc+V~d is the square root of the first binomial.") 2 149. Rudolff was convinced that development of a science is de- pendent upon its symbols. In the Preface to the second part of Rudolff 's Coss he states: "Das bezeugen alte bucher nit vor wenig jaren von der coss geschriben, in welchen die quantitetn, als dragma, res, substantia etc. nit durch character, sunder durch gantz geschribne wort dargegeben sein, vnd sunderlich in practicirung eincs yeden cxempels die frag gesetzt, ein ding, mit solchen worten, ponatur vna res." In translation: "This is evident from old books on algebra, written many years ago, in which quantities are represented, not by characters, but by words written out in full, 'drachm/ 'thing/ 'sub- stance/ etc., and in the solution of each special example the statement was put, 'one thing/ in such words as ponatur, una res, etc." 3 In another place Rudolff says: "Lernt die zalen der coss aus- sprechen vnnd durch ire charakter erkennen vnd schreiben." 4 ("Learn to pronounce the numbers of algebra and to recognize and write them by their characters.") DUTCH: GIELIS VAN DER HOECKE (1537) 150. An early Dutch algebra was published by Gielis van der Hoecke which appeared under the title, In arithmetica een sonderlinge excellet boeck (Antwerp [1537]) . 5 We see in this book the early appear- 1 Chr. Rudolff, Behend vnnd Hubsch Rechnung (Strassburg, 1525), fol. Rl". Quoted by Enestrom, Bibliotheca mathematica, Vol. XI (1910-11), p. 357. 2 Quoted from Rudolff by Enestrom, ibid., Vol. X (1909-10), p. 61. 3 Quoted by Gerhardt, op. tit. (1870), p. 153. This quotation is taken from the second part of Gerhardt's article; the first part appeared in the same publication, for the year 1867, p. 38-54. * Op. cit., Buch I, Kap. 5, Bl. Dijr; quoted by Tropfke, op. cit. (2. ed.), Vol. II, p. 7. 6 On the date of publication, see Enestrom, op. tit., Vol. VII (1906-7), p. 211; Vol. X (1900-10), p. 87. INDIVIDUAL WRITERS 137 ance of the plus and minus signs in Holland. As the symbols for powers one finds here the notation of Grammateus, N., pri., se., 3 a , 4 a , 5 a , etc., though occasionally, to fill out a space on a line, one en- " f I If fffftl PI* X ' ii mctiofotmt jAo tolt * ^^^^^^^^^^l 1 A jaL-j^Aty^'A'.y ''!, ^J^J'-'K'i'l 'U'^L':,*,1 ;/ 1' "'*' ^ 9 ^k^^it^^4mmm^mmm IT ,?, ' {,,f,VL . . - '( 1JT .J, i ' -*' ,..:' f ^r'S*^*' '^-''.-I'r *i,!Mlyd ' r ^ fni " * irr . ,., ' ' !l ' V ' '1 -*. ^;, ,^^ >^ r ^^ iifriftw^a^wt.^ffe^i^j^-^Mit tt^M^ ; V&'r-.J ' ' "i ' '.< :'' ;: ''; ; ,, ! ',.-'' '"';: ' H:; v S) ; i = r- :, ,_^';.:-: ; ;^.'|a| 9 ba,: :"-; , --^ -^ :^. FIG. 60. From Gielis van der Hoccke's In arithmetica (1537). Multiplica- tion of fractions by regule cos. counters numerus, num., or nu. in place of N.; also secu. in place of se. For pri. he uses a few times p. The translation of matter shown in Figure 60 is as follows: "[In order to multiply fractions simply multiply numerators by numera- 138 A HISTORY OF MATHEMATICAL NOTATIONS tors] and denominators by denominators. Thus, if you wish to multi- ply -j- by j > y u multiply 3x by 3, this gives 9#, which you write down. Then multiply 4 by 2x 2 , this gives 8x 2 , which you write under 9# . 9 the other ^ 2 . Simplified this becomes ^- , the product. Second rule: oX oX If you wish to multiply =- by Q 10 , multiply 20 by 16 [sic] which "" FIG. 61. Part of a page from M. Stifel's Arithmctica intcyra (1544), fol. 235 gives 320#, then multiply 2x by 3z+12, which gives 6x 2 +24x. Place this under the other obtained above a ^ 9 , nA , this simplified gives: 1 16 , the product. 3x 2 +12x As radical sign Gielis van der Hoecke does not use the German symbols of Rudolff, but the capital H of the Italians. Thus he writes (fol. 90B) "6+#8" for Q+l/S,-"-& 32 pri." for -l/32z. 1 The numerator should be 160, the denominator 3z-f-12. INDIVIDUAL WRITERS 139 GERMAN: MICHAEL STIFEL (1544, 1545, 1553) 151. Figure 61 is part of a page from Michael Stifel's important work on algebra, the Arithmetica Integra (Nurnberg, 1544). From the ninth and the tenth lines of the text it will be seen that he uses the same symbols as Rudolff had used to designate powers, up to and in- cluding x 9 . But Stifel carries here the notation as high as a; 16 . As Tropfke remarks, 1 the b in the symbol 6/3 of the seventh power leads Stifel to the happy thought of continuing the series as far as one may choose. Following the alphabet, his Arithmetica integra (1544) gives cp = x 11 , d@ = x 1 *, efl^x 11 , etc.; in the revised Coss of Rudolff (1553), Stifel writes 93/9, 0, J)0, (#0. He was the first 2 who in print dis- carded the symbol for dragma and wrote a given number by itself. Where Rudolff, in his Coss of 1525 wrote 4<, Stifel, in his 1553 edition of that book, wrote simply 4. A multiplication from Stifel (Arithmetica integra, fol. 236i>) 8 fol- lows: [Concluding part "63 + 87 6 of a problem:] 2$ 4 In Modern Symbols Go; 2 + 8x- 6 2x*- 4 We give Stifel's treatment of the quartic equation, 63+57+6 aequ. 5550: "Quaeritur numerus ad quern additum suum quadratum faciat 5550. Pone igitur quod quadratum illud faciat %. tune radix eius quadrata fac.iet IA. Et sic IA&+1A. aequabitur 1 Op. tit., Vol. II, p. 120. 2 Rudolff, Coss (1525), Signatur Hiiij (Stifel ed. [1553], p. 149); see Tropfke, op. tit., Vol. II, p. 119, n. 651. 3 Treutlein, op. tit., p. 39. 140 A HISTORY OF MATHEMATICAL NOTATIONS 5550. Itacq IA%. aequabit 5550 IA. Facit 1A. 74. Ergo cum. 2cC+6j+57+6, aequetur. 5550. Sequitur quod. 74. aequetur l3 + l#+2 ..... Facit itacq. IT^.8." 1 Translation: = 5,550 . Required the number which, when its square is added to it, gives 5,550. Accordingly, take the square, which it makes, to be A 2 . Then the square root of that square is A. Then A 2 +A= 5,550 and A 2 = 5,550-A. A becomes 74. Hence, since z 4 +2z 3 +6z 2 +5z+6 = 5,550, it follows that 74 = lz 2 +o;+2 ..... Therefore x becomes 8." 152. When Stifel uses more than the one unknown quantity 7> he at first follows Cardan in using the symbol q (abbreviation for quantita)* but later he represents the other unknown quantities by A, B, C ..... In the last example in the book he employs five un- knowns, 76? A, B, C, D. In the example solved in Figure 62 he repre- sents the unknowns by 7, A, B. The translation is as follows: "Required three numbers in continued proportion such that the multiplication of the [sum of] the two extremes and the difference by which the extremes exceed the middle number gives 4,335. And the multiplications of that same difference and the sum of all three gives 6,069. A-\-x is the sum of the extremes, A x the middle number, 2 A the sum of all three, 2x the difference by which the extremes exceed the middle. Then 2x multiplied into the sum of the extremes, i.e., in A+x, yields 2xA+2x 2 = 4,335. Then 2z multiplied into 2 A or the sum of all make 4xA = 6,069. "Take these two equations together. From the first it follows _ that xA = -^ - . But from the second it follows that IxA = Li 6,069 4,335 -2s 2 6,069 , . ., , , , _j _ t Hence - J - ^ - = I~~ > * or > Slnce ^ c y are e Q lia l to one and the same, they are equal to each other. Therefore [by reduction] 17,340 -8z 2 = 12, 138, which gives x 2 = 650^ and x = 1 Arithmetica Integra, fol. 307 B. 2 Ibid., Ill, vi, 252A. This reference is taken from H. Bosnians, Bibliotheca mathematica (3d ser., 1906-7), Vol. VII, p. 66. 141 ini IWi4'* jjttrt$t,jiffytt ^tf^,q*JWi tJUftpfjtttjr. ^ap/^* ->.' ^%^'X' 1 ; 1 ',:! 1 , i i inW, f| f^*'r tM^'l i ,V H - ~ "' i (| *| J Ju^i^mulr -u : f - l c^ ' - ,- .. , . FIG. 62. From Stifel's Ariihmelica Megra (1544), fol. 313 142 A HISTORY OF MATHEMATICAL NOTATIONS "It remains to find also \A. One has [as we saw just above] IxA r\ f -- . Since these two are equal to each other, divide each by x, and r* f there follows -4 = -7 . But as x = 25|, one has 4x=102, and 6,069 ~ divided by 102 gives 59|. And that is what A amounts to. Since AXJ i.e., the middle number equals 34, and A +x, i.e., the sum of the two extremes is 85, there arises this new problem : "Divide 85 into two parts so that 34 is a mean proportional between them. These are the numbers: B, 34, 85-JS. Since 85B-B 2 = 1,156, there follows #=17. And the numbers of the example are 17, 34, 68." Observe the absence of a sign of equality in Stifel, equality being expressed in words or by juxtaposition of the expressions that are equal; observe also the designation of the square of the unknown B by the sign B%. Notice that the fractional line is very short in the case of fractions with binomial (or polynomial) numerators a singularity found in other parts of the Arithmetica integra. Another oddity is Stifel's designation of the multiplication of fractions. 1 They are writ- ten as we write ascending continued fractions. Thus *** means "Tres quartae, duarum tertiarum, uriius septimae," i.e., of f off The example in Fig. 62 is taken from the closing part of the Arith- metica integra where Cardan's A rs magna, particularly the solutions of cubic and quartic equations, receive attention. Of interest is St if el's suggestion to his readers that, in studying Cardan's Ars magna, they should translate Cardan's algebraic statements into the German symbolic language: "Get accustomed to transform the signs used by him into our own. Although his signs are the older, ours are the more commodious, at least according to my judgment." 2 1 Arithmetica integra (1548), p. 7; quoted by S. Giinther, Vermischte Unter- suchungen (Leipzig, 1876), p. 131. 2 Arithmetica integra (Niirnberg, 1544), Appendix, p. ,306. The passage, as quoted by Tropfke, op. cit., Vol. II (2. ed.), p. 7, is as follows: "Assuescas, signa eius, quibus ipse utitur, transfigurare ad signa nostra. Quamvis enim signa quibus ipse utitur, uetustiora sint nostris, tainen nostra signa (meo quids iudicio) illis sunt commodiora." INDIVIDUAL WRITERS 143 153. Stifel rejected RudolfTs symbols for radicals of higher order and wrote j/j for i/~~, \/cC for f~~, etc., as will be seen more fully later. But he adopts Rudolff s dot notation for indicating the root of a binomial r 1 1/3-12-1/56 has for its square 12+1/^6+12- l/i6-i/g!38-i/jl38"; i.e., "V\2+ 1/6-1/12 -1/6 has for its square 12 +l/6> 12 -1/6 -1/138 -1/138." Again: 2 "Tcrtio vide, utru i/Vi/3 12500-50 addita ad i/j-j/J 12500+50. faciat j/V 1/350000+200" ("Third, see whether ^1/12,500-50 added to ^1/12,500+50 makes 1/1/50,000+200"). The dot is employed to indicate that the root of all the terms following is required. 154. Apparently with the aim of popularizing algebra in Germany by giving an exposition of it in the German language, Stifel wrote in 1545 his Deutsche arithmetical in which the unknown x is expressed by sum, x 2 by "sum: sum," etc. The nature of the book is indicated by the following equation: "Der Algorithmic meiner deutschen Coss branch t zurn ersten schlecht vnd ledige zale | wie der gemein Algorithmus | als da sind 12345 etc. Zuin audern braucht er die selbigen zalen vnder diesern namen | Suma. Vnd wirt dieser nam Suma | also verzeichnet | Sum : Als hie I 1 sum: 2 sum: 3 sum etc ..... So ich aber 2 sum: Multi- plicir mit 3 sum : so komen rnir 6 sum: sum: Das mag ich also lesen 6 summe summarum | wie man den im Deutsche offt findet [ suma sumarum ..... Soil ichmultipliciren6sum: sum: sum: mit 12 sum: sum: sum: So sprich ich | 12 mal 6. macht 72 sum: sum: sum: sum: sum sum . . . ," 4 Translation: "The algorithm of my Deutsche Coss uses, to start with, simply the pure numbers of the ordinary algorithm, namely, 1, 2, 3, 4, 5, etc. Besides this it uses these same numbers under the name of summa. And this name summa is marked sum:, as in 1 sum: 2 sum: 3 sum, etc ..... But when I multiply 2 sum: by 3 sum: I obtain 6 sum: sum:. This I may read | 6 summe summarum \ for in German one encounters often suma sumarum ..... When I am to multiply 6 sum: sum: sum: by 12 sum: sum: sum:, I say | 12 times 6 makes 72 sum: sum: sum: sum: sum: sum: . . . ." 1 Op. cit., fol. 138a. 2 Ibid., fol. 315a. 3 Op. oil. Inhaltend. Die Hauszrechnung. Deutsche Coss. Rechnung (1545). 4 Treutlein, op. cit., Vol. II, p. 34. For a facsimile reproduction of a page of Stifel's Deutsche arithmetica, see D. E. Smith, Rara arithmetica (1898), p. 234. 144 A HISTORY OF MATHEMATICAL NOTATIONS The inelegance of this notation results from an effort to render the subject easy; Stifel abandoned the notation in his later publications, except that the repetition of factors to denote powers reappears in 1553 in his "Cossische Progress" ( 156). In this work of 1545 Stifel does not use the radical signs found in his Arithmetica integra; now he uses -%_/, /, I/, for square, cube, and fourth root, respectively. He gives (fol. 74) the German capital letter 2K as the sign of multiplication, arid the capital letter 2) as the sign of division, but does not use either in the entire book. 1 155. In 1553 Stifel brought out a revised edition of RudolfFs Coss. Interesting is StifePs comparison of RudoliTs notation of radicals with his own, as given at the end of page 134 (see Fig. 63a), and his declaration of superiority of his own symbols. On page 135 we read: "How much more convenient my own signs are than those of Rudolff, no doubt everyone who deals with these algorithms will notice for himself. But I too shall often use the sign i/ in place of the 1/j, for brevity. "But if one places this sign before a simple number which has not the root which the sign indicates, then from that simple number arises a surd number. "Now my signs are much more convenient and clearer than those of Christoff . They are also more complete for they embrace all sorts of numbers in the arithmetic of surds. They are [here he gives the symbols in the middle of p. 135, shown in Fig. 636]. Such a list of surd numbers ChristofFs symbols do not supply, yet they belong to this topic. "Thus my signs are adapted to advance the subject by putting in place of so many algorithms a single and correct algorithm, as we shall see. "In the first place, the signs (as listed) themselves indicate to you how you are to name or pronounce the surds. Thus, j//36 means the sursolid root of 6, etc. Moreover, they show you how they are to be reduced, by which reduction the declared unification of many (indeed all such) algorithms arises and is established/' 156. Stifel suggests on folio 61# also another notation (which, however, he does not use) for the progression of powers of x t which he calls "die Cossische Progress" We quote the following: "Es mag aber die Cossische Progress auch also verzeychnet wer- den: 012 3 4 1 \A IAA IAAA IAAAA etc. 1 Cantor, op. tit,, Vol. II (2. cd., 1913), p. 444. INDIVIDUAL WRITERS 145 auch also: 0123 4 M# IBB IBBB IBBBB etc. Item auch also: 0123 4 1 - 1C ICC - 1CCC 1CCCC etc. Vnd so fort an von andern Buchstaben." 1 . , FIG. 63a.~ This shows p. 134 of StifeTs edition of Rudolffs Coss (1553) 1 Treutlein, op. cit., Vol. II (1879), p. 34. 146 A HISTORY OF MATHEMATICAL NOTATIONS We see here introduced the idea of repeating a letter to designate powers, an idea carried out extensively by Harriot about seventy-five /jgrufnicKrfHftivof i. out FIG. 636, This shows p. 135 of Stifel's edition of Rudolff s Coss (1553) INDIVIDUAL WRITERS 147 years later. The product of two quantities, of which each is repre- sented by a letter, is designated by juxtaposition. GERMAN: NICOLAUS COPERNICUS (1566) 157. Copernicus died in 1543. The quotation from his De revolu- tionibus orbium coelestium (1566; 1st ed., 1543) 1 shows that the exposi- tion is devoid of algebraic symbols and is almost wholly rhetorical. We find a curious mixture of modes of expressing numbers: Roman numerals, Hindu-Arabic numerals, and numbers written out in words. We quote from folio 12: "Circulum autem communi Mathematicorum consensu in CCCLX. partes distribuirnus. Dimetientem uero CXX. partibus asciscebant prisci. At posteriores, ut scrupulorurn euitarent inuolu- tionem in multiplicationibus & diuisionibus numcrorum circa ipsas lincas, quae ut plurimum incommensurabiles sunt longitudine, saepius etiam potentia, alij duodccies centena milia, alij uigesies, alij aliter rationalem constituerunt diametrum, ab eo tern pore quo indicae numerorurn figurae sunt usu receptae. Qui quidem numerus qucm- cunque alium, sine Graccum, sine Latinum singular! quadam prompti- tudine superat, & omni generi supputationum aptissime sese accommo- dat. Nos quoq, earn ob causam accepimus diametri 200000. partes tanquam sufficientes, que, possint errorern excludere paten tern." Copernicus does not seem to have been exposed to the early move- ments in the fields of algebra and symbolic trigonometry. GERMAN: JOHANNES SCHEUBEL (1545, 1551) 158. Scheubel was professor at the University of Tubingen, and was a follower of Stifcl, though deviating somewhat from Stifcl's notations. In ScheubeFs arithmetic 2 of 1545 one finds the scratch method in division of numbers. The book is of interest because it docs not use the + and signs which the author used in his algebra; the + and were at that time not supposed to belong to arithmetic proper, as distinguished from algebra. 1 Nicolai Copcrnici Torinensis de Revolvlionibus Orbium Codcstium, Libri VI. .... Item, de Libris Revolvtionvm Nicolai Copcrnici Narratio prima, per M. Georgi- um loachimum Rheticum ad D. loan. Schonerum scripla. Basileae (date at the end of volume, M.D.LXVI). 2 De Nvmeris el Diversis Rationibvs seu Regulis computalionum Opusculum, a loanne Schcubelio compositum .... (1545). 148 . A HISTORY OF MATHEMATICAL NOTATIONS Scheubel in 1550 brought out at Basel an edition of the first six books of Euclid which contains as an introduction an exposition of algebra, 1 covering seventy-six pages, which is applied to the working of examples illustrating geometric theorems in Euclid. 159. Scheubel begins with the explanation of the symbols for powers employed by Rudolff and Stifel, but unlike Stifel he retains a symbol for numerus or dragma. He explains these symbols, up to the twelfth power, and remarks that the list may be continued indefinitely. But there is no need, he says, of extending this unwieldy designation, since the ordinal natural numbers afford an easy nomenclature. Then he introduces an idea found in Chuquet, Grammateus, and others, but does it in a less happy manner than did his predecessors. But first let us quote from his text. After having explained the symbol for dragma and for x he says (p. 2) : "The third of them 3, which, since it is produced by multiplication of the radix into itself, and indeed the first [multiplication], is called the Prima quantity and furthermore is noted by the syllable Pri. Even so the fourth c, since it is produced secondly by the multiplication of that same radix by the square, i.e., by the Prima quantity, is called the Second quantity, marked by the sylla- ble Se. Thus the fifth sign 33, which springs thirdly from the multiplica- tion of the radix, is called the Tertia quantity, noted by the syllable Ter " 2 And so he introduces the series of symbols, N. t Ra., Pri., Re., Ter., Quar., Quin., Sex., Sep , which are abbreviations for the words numerus, radix, prima quantitas (because it arises from one multiplication), secunda quantitas (because it arises from two multi- plications), and so on. This scheme gives rise to the oddity of desig- nating x n by the number n I, such as we have not hitherto encoun- tered. In Pacioli one finds the contrary relation, i.e., the designation of x n ~ l by x n ( 136). ScheubcPs notation does not coincide with that of Grammateus, who more judiciously had used pri., se., etc., to desig- nate X, x 2 , etc. ( 147). ScheubePs singular notation is illustrated by 1 Evdidis Megarensis, Philosophi et Mathematici excellentissimi, sex libri prior es de Geometrids principijs, Graed et Latini .... Alyebrae porro rcgvlae, propter nvmerorum excmpla, passim proposition/thus adiecta, his libris praemissae sunt, eadenque demonstratae. Author e loanne jSchcvbelio, .... Basileae (1550). I used the copy belonging to the Library of the University of Michigan. 2 "Tertius de, g. qui cu ex multiplicatione radicis in se producatur, et primo quidem: Prima quantitas, et Pri etiam syllaba notata, appclletur. Quartus uer6 cC quia ex multiplicatione ciusdem radicis cum quadrato, hoc eat, cum prima quanti- tate, secundd producitur: Se syllaba notata, Secunda quantitas dicitur. Sic character quintus, gg, quia ex multiplicatione radicis cum secunda quantitate tertio nascitur: Ter syllaba notata, Tertia etiam quantitias dicitur " INDIVIDUAL WRITERS 149 Figure 64, where he shows the three rules for solving quadratic equa- tions. The first rule deals with the solution of 4x 2 +3x = 217, the sec- ond with 3x+175-4o; 2 , the third with 3z 2 +217 = 52z. These differ- ent cases arose from the consideration of algebraic signs, it being de- sired that the terms be so written as to appear in the positive form. Only positive roots are found. ALIVD EXEMPLVM. P R I M I C A N O N I $. $ E C V N D I C A N O N 1 $. Prf. ra, N ra, N prf. 4 -H J gquales 217 3 + 17? arqu. 4 Hie, quia maximi charafleris nttmerus non cftumtasjdiuifion^utdidhrai" eft, ci fuccuni debct. Venfunt autem fa<3a diuifionc, pru ra N ra N pnb | in fe, f ? ucnu ^||. funt 7 f minus f . font <j| plus | manent 7 ucmunt 7 radios ualon radicfs ualor. ALIVD TERT1I CANONU a pri, H- 217 N acquales 51 ra. Ethic,qma maximi characflens numerusnoncftunitas,diuifioncci(uccu^ rcndum erit. Veniunc autem hoc fadto, t pri> -t~ a | 7 N requales f f N J f infe. * 2 /> minus I j 7 ,manec ^ fde Hufus ra. qua, eft tf < sf, &Tmanent7,uelproue* tad niunt 10 1, Vtercp radio's ualor , quod examinari poteft FIG. 64. 'Part of p. 28 in Sehcubel's Introduction to his Euclid, printed at Basel in 1550. Under proportion we quote one example (p. 41) : " 3 ra.+4 N. ualent 8 se.+4: pri. quanti 8 ter.4: ra. 64 sex. +32 quin. -32 ter. - 16 se. Foot - ~ - " 150 A HISTORY OF MATHEMATICAL NOTATIONS In modern notation: 3z+4 are worth &r 3 +4;r 2 how much 8x 4 4x . 64z 7 +32z 6 -32o; 4 - Wx* Result - . In the treatment of irrationals or numeri surdi Scheubel uses two notations, one of which is the abbreviation Ra. or ra. for radix, or "square root," ra.cu. for "cube root," ra.ra. for "fourth root." Con- fusion from the double use of ra. (to signify "root" and also to signify x) is avoided by the following implied understanding: If ra. is fol- lowed by a number, the square root of that number is meant; if ra. is preceded by a number, then ra. stands for x. Thus "8 ra." means Sx; "ra. 12" means 1/12. Scheubel's second mode of indicating roots is by RiidolfTs sym- bols for square, cube, and fourth roots. He makes the following state- ment (p. 35) which relates to the origin of j/: "Many, however, are in the habit, as well they may, to note the desired roots by their points with a stroke ascending on the right side, and thus they prefix for the square root, where it is needed for any number, the sign j/: for the cube root, AW/ ; and for the fourth root AA/-" 1 Both systems of notation are used, sometimes even in the same example. Thus, he considers (p. 37) the addition of "ra. 15 ad ra. 17" (i.e., 1/15+1/17) and gives the result "ra.coZ. 32+1/1020" (i.e. 1/32 +1/1, 020). The ra.col. (radix collecti) indicates the square root of the binomial. Scheubel uses also the ra.re (radix residui) and radix binomij. For example (p. 55), he writes "ra.re. i/15-i/12" for 1/1/15-1/12. Scheubel suggests a third notation for irrationals (p. 35), of which he makes no further use, namely, radix se. for "cube root," the abbrevia- tion for secundae quantitatis radix. The algebraic part of ScheubePs book of 1550 was reprinted in 1551 in Paris, under the title Algebrae compendiosa facilisqve description 1 "Solent tamen multi, ct benc etiam, has desideratas radices, suis punctis cum lines quadam a dextro latere asccndente, notare, atque sic pro radice quidern quadrata, ubi haec in aliquo nurnero dosideratur, not am \/\ pro cubica uero, /VV\/ : ac radicis radice deinde, /w/ praeponunt." 2 Our information on the 1551 publication is drawn from H. Staigmtiller, "Johannes Scheubel, ein deutscher Algebraiker des XVI. Jahrhunderts," Abhand- lungen zur Geschichte der Mathematik, Vol. IX (Leipzig, 1899), p. 431-69; A. Witting and M. Gebhardt, Beispielc zur Geschichte der Mathematik, II. Teil INDIVIDUAL WRITERS 151 It is of importance as representing the first appearance in France of the symbols + and and of some other German symbols in algebra. Charles Hutton says of Scheubel's Algebrae compendiosa (1551): "The work is most beautifully printed, and is a very clear though succinct treatise; and both in the form and matter much resembles a modern printed book." 1 MALTESE: WIL. KLEBITIUS (1565) 160. Through the courtesy of Professor H. Bosnians, of Brussels, we are able to reproduce a page of a rare and curious little volume containing exercises on equations of the first degree in one unknown number, written by Wilhelrn Klebitius and printed at Antwerp in 1565. 2 The symbolism follows Scheubel, particularly in the fancy form given to the plus sign. The unknown is represented by "1R." The first problem in Figure 65 is as follows: Find a number whose double is as much below 30,000 as the number itself is below 20,000. In the solution of the second and third problems the notational peculi- arity is that J/2. -J- is taken to mean Jfi. -J-/i., and 1/J.- to mean IR.-iR. GERMAN: CIIRISTOPHORUS CLAVIUS (1608) 161. Though German, Christophorus Clavius spent the latter part of his life in Rome and was active in the reform of the calendar. His Algebra? marks the appearance in Italy of the German + and signs, and of algebraic symbols used by Stifel. Clavius is one of the very first to use round parentheses to express aggregation. From his Algebra we quote (p. 15): "Pleriqve auctores pro signo + ponunt literam F, vt significet plus: pro signo vero ponunt literam M, vt significet minus. Bed placet nobis vti nostris signis, vt a literis distinguantur, ne confusio oriatur." Translation: "Many authors put in place of the sign + the letter P, which signifies "plus": (Leipzig-Berlin, 1913), p. 25; Tropfke, op. cit., Vol. I (1902), p. 195, 198; Charles Hutton, Tracts on Mathematical and Philosophical Subjects, Vol. II (London, 1812), p. 241-43; L. C. Karpinski, Robert of Chester's .... Al-Khowarizmi, p. 39-41. 1 Charles Hutton, op. cit., p. 242. 2 The title is Insvlae Melitensis, qvam alias Maltam vocant, Historia, quaestionib. aliquot Mathematicis reddita iucundior. At the bottom of the last page: "Avth. Wil. Kebitio." * Algebra Christophori Clavii Bambergensis e Societate lesv. (Romae, M.DC.VIII). 152 A HISTORY OF MATHEMATICAL NOTATIONS likewise, for the sign they put the letter M f which signifies "minus." But we prefer to use our signs; as they are different from letters, no confusion arises/' In his arithmetic, Clavius has a distinct notation for "fractions of fractional numbers," but strangely he does not use it in the ordinary .:t. ad H;^*p '" rr :"' ^ ,.:,,;t:*.*l'|, ' '" ' FIG. 65.- Page from W. Klebitius (1565) multiplication of fractions. His means $ of 4 He says: "Vt praedicta minutia minutiae ita scribenda est f \ pronuntiaturque sic. Tres quintae quatuor septimaru vnius integri." 1 Similarly, i 2 * a ' 1 * yields T fy. The distinctive feature in this notation is the 1 Epitome arithmeticae (Rome, 1583), p. 68; see also p. 87. INDIVIDUAL WRITERS 153 omission of the fractional line after the first fraction. 1 The dot cannot be considered here as the symbol of multiplication. No matter what the operation may be, all numbers, fractional or integral, in the C A P. XXVIII. ,5* SferurfusBinomtumprimum 7^^2880, Maius nomcnya. fecabitur in duas partes producentes 710* quartam partem quadrati 1880. maioris nominis , hac ratione* Semi/Hsmaiorisnomini*7a-eil36. a / 60 + J% 12 cuius quadratoxij^/detraSaquarti Jfr 60 -j- J% n pars pratdi&a 7*0* relinquft T7*. cur , , v _ rt , t _ J ins radix 24* addita ad femiffem uo- 6 T jj T " minaram 3 6. & detrafta a.b eadein/a* /"^ ff ..'.?. _ . c it parces ^ux/Ttas ^o, & x s. Ergo ra- 721 Hh Vy 18.80 dix Binomij eft JK 6o-|*^ n. cjuod hie probatum eft per muitiplicationem radicis in (e quadrate Sit quoque el icienda radix exhoc refiduo fcxto JK 60 Vfr 12; Matus nomen Vj< 6o.diflribuetar in duas partes producetes^.quar-* tarn parrem quadrat: i *. minoris nominis r hoc pa^to Semiflis ma* ions nominis 4% 60+ eft J% ij.,a emus quadrato ij.detraCla nomi- nata pars quarta 5. relinquit u. cuius radix J# n^ addita ad fe- miirem J% i j. prardi<flam, 5: ab ? adem fublata facit partes JK i f -f* VK 1 2. & J% i $ Vfr i a. Ergo radi!x difli Re/Tduf fexti eft 7 i J + J% i i>* ^ (Vjf i J Vfc Tz) quod He probatum eft * if + Vjf t*) ~ ^ fJK IT ft Ji 15 +</* u; Jy 6/y i? ~ ^/y n Quadrata partium. J% ij + <!% ** & Summa. *1% 6* Jjf. ix Nam quadrata pattium faciunt Jv 6*. nimirum* duplum J% i j. Et ex vna parte Jlf (J% i J 4 Vtf i *J in alteram 7- /^y i y J% n) fit <$?. quippecumqiradratumix ex quadrato ij.fubduftum relinquat 3. cm prarponendum eft fignum /^ cum ffgno . pro* ter Refiduum. Duplum autem-r 1 ^ff J* tacit ^/K u^ FIG. 66. A page in Clavius' Algebra (Rome, 1608). It shows one of the very earliest uses of round parentheses to express aggregation of terms. arithmetic of Clavius are followed by a dot. The dot made the numbers stand out more conspicuously. 1 In the edition of the arithmetic of Clavius that appeared at Cologne in 1601, p. 88, 126, none of the fractional lines are omitted in the foregoing passages. 154 A HISTORY OF MATHEMATICAL NOTATIONS As symbol of the unknown quantity Clavius uses 1 the German 7 In case of additional unknowns, he adopts IA, IB, etc., but he refers to the notation lq, 2q, etc., as having been used by Cardan, Nonius, and others, to represent unknowns. He writes: 33^+4^4, 4B3A for Clavius' Astrolabium (Rome, 1593) and his edition of the last nine books of Euclid (Rome, 1589) contain no algebraic symbolism and are rhetorical in exposition. BELGIUM: SIMON STEVIN (1585) 162. Stevin was influenced in his notation of powers by Bombelli, whose exponent placed in a circular arc became with Stevin an ex- ponent inside of a circle. Stevin's systematic development of decimal fractions is published in 1585 in a Flemish booklet, La thiende,' 2 and also in French in his La disme. In decimal fractions his exponents may be interpreted as having the base one-tenth. Page 16 (in Fig. 67) shows the notation of decimal fractions and the multiplication of 32.57 by 89.46, yielding the product 2913.7122. The translation is as follows: "III. Proposition, on multiplication: Being given a decimal frac- tion to be multiplied, and the multiplier, to find their product. "Explanation of what is given: Let the number to be multiplied be 32.57, and the multiplier 89.46. Required, to find their product. Process: One places the given numbers in order as shown here and multiplies according to the ordinary procedure in the multiplication of integral numbers, in this wise: [see the multiplication]. "Given the product (by the third problem of our Arithmetic) 29137122; now to know what this means, one adds the two last of the given signs, one (2) and the other (2), which are together (4). We say therefore that the sign of the last character of the product is (4), the which being known, all the others are marked according to their successive positions, in such a manner that 2913.7122 is the required product. Proof: The given number to be multiplied 32.57 (according to the third definition) is equal to 32^ ^fa, together 32-^fo. And for the same reason the multiplier 89.46 becomes 89-^oV Multiplying the said 32jVo by the same, gives a product (by the twelfth problem of our Arithmetic) 2913iVoVoJ but this same value has also the said product 2913.7122; this is therefore the correct product, which we 1 Algebra, p. 72. 2 A facsimile edition of La "thiende" was brought out in 1924 at Anvers by H. Bosnians. INDIVIDUAL WRITERS 155 were to prove. But let us give also the reason why multiplied by 0, gives the product (which is the sum of their numbers), also why times gives the product , and why (O) times gives , etc. We take -& and T $ T (which by the third definition of this Disme s* are .2 and .03; their product is which, according to our third 1UUU definition, is equal to .006. Multiplying, therefore, by gives the Iff, 1,-f, 4^ i ~ r ,* . _.* 'i.rJ5Jl' ' . L /'; [ M,J FIG. 67. Two pages in S. Stevin's Thiende (1585). The same, in French, is found in Les ceuvres mathematiques de Simon Stevin (ed. A. Girard; Leyden, 1634), p. 209. product , a number made up of the sum of the numbers of the given signs. Conclusion: Being therefore given a decimal number as a multiplicand, and also a multiplier, we have found their product, as was to be done. "Note: If the last sign of the numbers to be multiplied is not the same as the sign of the last number of the multiplier, if, for example, the one is 30708, and the other 5040, one proceeds as above 156 A HISTORY OF MATHEMATICAL NOTATIONS and the disposition of the characters in the operation is as shown: [see process on p. 17]." A translation of the La disme into English was brought out by Robert Norman at London in 1608 under the title, Disme: The Art of QVSSTION XX* * un@td, qucfonquarre it muttipMptrU fotnmt du double d icdui, &lcquarrcde i # 4, It CoKSr&VCTION. Soitlenombrerequis 10 4 Son qUarrc i @, auquelajouftc it Qui mulripliepar la fpmmp du double du nota- Jbre rcquis , & le quarr^ dc z c 4 > qtii eft Bgalau quarrc du produidde- i,Dar id , Lcfqucls rcduiiis, i (era cgale a i 48 ; Et i par Ic 71 pf oblcmc, vaudra 4. Jc di quc 4 eft Ic nombrt f cquis. Demon ftration. Le quarrc dc 4*eft 16, qui ivcc u fai6l 4 , qui mulripli6 par i (i tf pour la fomme du double d'iceluy 4, & Ic. quarrc de i &: encore 4 ) faift ^4 , qui font egales ail quarredu produiddc i,par Ie4trouve ? felon le re- qu'il falloit deoioourer* FIG. 68. From p. 98 of U arithmetique in Stevin's (Euvres mathematiqucs (I^eyden, 1634). Tenths, or Dedmall Arithmetike. Norman does not use circles, but round parentheses placed close together, the exponent is placed high, as in ( 2 ). The use of parentheses instead of circles was doubtless typographically more convenient. Stevin uses the circles containing numerals also in algebra. Thus INDIVIDUAL WRITERS 157 a circle with 1 inside means x, with 2 inside means x 2 , and so on. In Stevin's (Euvres of 1634 the use of the circle is not always adhered to. Occasionally one finds, for x 4 , for example, 1 the signs (4) and (4). The translation of Figure 68 is as follows: "To find a number such that if its square 12, is multiplied by the sum of double that num- ber and the square of 2 or 4, the product shall be equal to the square of the product of 2 and the required number. Solution "Let the required number be x 4 Its square x 2 , to which is added 12 gives x 2 12 4 This multiplied by the sum of double the re- quired number and the square of 2 or 4, i.e., by 2x+8, gives 2x 3 +8x 2 - 24x- 96 equal to the 64 square of the product pf 2 and x, i.e., equal to. ... 4x 2 Which reduced, x 3 = - 2x 2 +12x+48; and x, by the problem 71, becomes 4. I say that 4 is the required number. "Demonstration: The square of 4 is 16, which added to 12 gives 4, which multiplied by 16 (16 being the sum of double itself 4, and the square of 2 or 4) gives 64, which is equal to the square of the product of 2 and 4, as required; which was to be demonstrated. " If more than one unknown occurs, Stevin marks 2 the first un- known "1O," the second "1 secund. O," and so on. In solving a Diophantine problem on the division of 80 into three parts, Stevin represents the first part by "1O," the second by "1 secund. O," the third by "- -1 secund. +80." The second plus J the first + 6 minus the binomial | the second + 7 yields him " secund. O-f -JO 1." The sum of the third and J the second, + 7, minus the binomial \ the third + 8 yields him "|O H secund. O + 4 f a -" By the conditions of the problem, the two results are equal, and he ob- tains "1 Secund. O Aequalem {J JO +45." In his U arithmetique? one finds "12 sec. +23@M sec. +10," which means 12i/ 4 + 23x?/ 2 +10x 2 , the M signifying here "multiplication" as it had with Stifel ( 154). Stevin uses also D for "division." 163. For radicals Stevin uses symbols apparently suggested by 1 Les (Euvres malhematiques de Simon Stevin (1634), p. 83, 85. 2 Stevin, Tomvs Qvintvs mathematicorvm Hypomnemalvm de Miscellaneis (Leiden, 1608), p. 516. 8 Stevin, (Euvres mathtmatiques (Leyden, 1634), p. 60, 91, of "Le II. livre d'arith." 158 A HISTORY OF MATHEMATICAL NOTATIONS those of Christoff Rudolff, but not identical with them. Notice the shapes of the radicals in Figure 69. One stroke yields the usual square root symbol j/, two strokes indicate the fourth root, three strokes the eighth root, etc. Cube root is marked by y followed by a 3 inside a circle; VA/ followed by a 3 inside a circle means the cube root twice taken, i.e., the ninth root. Notice that i/^X means 1/3 times z 2 , not I/So; 2 ; the X is a sign of separation of factors. In place of the u or v to express "universal" root, Stevin uses bino ("binomial") root. Stevin says that f placed within a circle means #*, but he does not actually use this notation. His words are (p. 6 of (Euvres [Arithmetic]), "f en un circle seroit le charactere de racine quarre de (a), par ce que telle f en circle multiplide en soy donne produict (5), et ainsi des autres." A notation for fractional exponents had been suggested much earlier by Oresme (123). LORRAINE: ALBERT GIRARD (1629) 164. Girard 1 uses + and , but mentions -r as another sign used for "minus." He uses = for "difference entre les quantitez oft il se treuve." He introduces two new symbols: ff, plus que; , moins que. In further explanation he says : "Touchant les lettres de F Alphabet au lieu des nombres: soit A & aussi B deux grandeurs: la somme est A+B, leur difference est A=B, (ou bien si A est majeur on dira que .A c'est A B) leur produit est AB, mais divisant A par B viendra -~ comme s fractions: les voyelles se posent pour les choses incognues." This use of the vowels to represent the unknowns is in line with the practice of Vieta. The marks (2), (3), (4), . . . . , indicate the second, third, fourth, . . . . , powers. When placed before, or to the left, of a number, they signify the respective power [BRIEF VE COLLECTION DES of that number; when placed CHA-RACTERES QjfON vsERA B N| after a number, they signify CIST* AIUTHMETK^VE. t he power of the unknown -TTEuqucUcognoinincedcscharadlcrcscftacgran- quantity. In this respect V dcconrcqucncc,parcequonlufccnrArithiiic- . , , au lieu d mots, nous les ajoufterons icy, (com- uirard f ollows the general plan ^bicnquauprcccdcntchafcunacftcamplcmcntacciarc f oun d in Schoner's edition of [Continued on page i5'jj the Algebra of Ramus. But 1 Invention nouvelle en I'Algebre, A Amsterdam (M.DC.XXIX); reimpression par Dr. D. Bierens de Haan (Leiden, 1884), fol. B. INDIVIDUAL WRITERS 159 4 tcr Girard adopts the practice of Stevin in using fractional ex- ponents. Thus, "(f)49" means (1/49) 3 = 343, w hile "49(f)" means 49z 3 . He points out that 18(0) is the same as 18, that (1)18 is the same as 18(0). We see in Girard an ex- tension of the notations of Chuquet, Bombelli, and Ste- vin ; the notations of Bombelli and Stevin are only variants of that of Chuquet. The conflict between the notation of roots by the use of fractional exponents and by the use of radical signs had begun at the time of Girard. "Or pource que y est en usage, on le pourra prendre au lieu de (-0 a cause aussi de sa facilite, signifiant racine seconde, ou racine quaree; que si on veut pour- suivre la progression on pour- ra au lieu de j/ marquer i/ '; & pour la racine cubique, ou tierce, ainsi \/ ou bien (), ou bi6 cf, ce qui peut estre au choix, mais pour en dire mon opinion les fractions sont plus expresses & plus propres a exprimer en perfection, & j/ plus faciles et expedientes, comme i/ 32 est a dire la ra- cine de 32, & est 2. Quoy que ce soit Tun & 1'autre sont f acils a comprendre, mais |/ et c sont pris pour faciliteV' Girard appears to be the first to suggest placing the index of the root in the opening of the radical sign, as {/. Sometimes he writes j/V for \/. en h definition,) par ordrc rous cnfcmble c<3mc s'cnfuic. Lcscbiraderesligmrians qu3mitcz,dcfquel$ 1'cxpli- cation fc trouvc cs 14.15.16 ^7.18. definitions, font tcls. Coramcnctmcnt dc quantusi oui eft nombro Atith. . pu radical quclconquc.. prime quamite. (2, lecondcquanurc. (<} tierce quamitc. quartc qiiamirc ,ix c. Lcs ch.ua ctetc> iignili.uis poOpofccs qunnrircz, dcfquels 1'cxplication fc trouvc d Li 18 definition, font tcls: 1 icc,j Vnc prime quandti (ccondcmcncpofcc. j) C^unttc Tccondcs qiuntitcz tict'cemciu pofcci, ou pioccdans de la prime quantitc dctcc- mcnt pofcc. i (|j Tcc0 Protluicl d'une prime quantitc par unc prime 'qu.inrirc fccondeincntpofcc. 5 tcr(7}Pioduik de cincq quattesquantitez par itnc rccondcquintiteticrccmtnt pofcc. Lcs charadlcics n^nifians ndnc dc'quels {'expli- cation ft 1 trouvc a la ip (k 30 clctimuoa iont tcls : 4/ Ratine dc qu.irrc. Hi/ Racine de racine dc quarre. ' t*4/ -' Racine dc racine dc racine dc quarre. #uv^ R.ictnc (ic racine dc racine de racine dc quarre, 4/C<) Racine dc cube. <*/ (?) Racuic*de racine dc cube, 4/ (j)R;icinc dcquirrfe quantitc. 4</vi)R ac iocdc racine dc quane cjuantitc,&.'c. Lc characlcic /Iguifiant b feparation entic le fi- gne de racine & la quautitc, duqucl 1'cxplication ic trouvc i la 34. definition, cfbtcl. ^ X Comme i/ 3 X n ' c 't p ls le mcfinc q[ue </ 5 (7 1 , cominc did eft a lididtc 54. definition. Les chaia&cics Jlgnifiaiis plus <Sc moiiw, comme 1 la $6 dctmition, font tcls : Moins. Et pour cxpliquer la racine d'un multlnomic Cqu'aucuns arpcllcm vacinc univcricllc) nousule- rons le vocable da multinomic, comme: 4/bino i,ri~'<v $ c'eft a dire racine quarrccdc bino- mie,oudc laiommcdc z & 4/5. / tiino V ,3,-r V i V 5 , c'cft a dire Vacinc quarrc'c dc rrinomie , ou de la iommc de */ 3 &c */ i <?c !X'i^i-/i/ , c 'eft a dire racine cubique de c'cftadifc racine quarrcc dcbino- j */fein^ binonne /l/bino ; f bino i Qf; -j- 1 CD> c'eft a dire racine cubique dc bi notnic i (1^, -f- 1 0, &c. FIG. 69. From S. Stevin ; s L'arithmttique in (Euvres math&matiques (ed. A. Girard; Leyden, 1634), p. 19. 160 A HISTORY OF MATHEMATICAL NOTATIONS The book contains other notations which are not specially ex- plained. Thus the cube of B+C is given in the form B(B q +C 3 Q ) + C(Bl+C q ). We see here the use of round parentheses, which we encountered before in the Algebra of Clavius and, once, in Cardan. Notice also that C\ means here 3C 2 . Autre exemple In Modern Symbols "Soit 1(3) esgale & -G(l)+20 Let x = -6z+20 Divisons tout par 1(1) Divide all by x, 20 90 1(2) esgale a -6+ ." x 2 = -6+ . Again (fol. F3) : "Soit 1(3) esgale a 12(1) - 18 (impossible d'estre esgal) car le ^ est 4 9 qui est \ do 18 son cube 64 81 son quarr . Et puis que 81 est plus que 64, Pequation est im- possible & inepte." Translation: "Let 0?= 12z 18 (impossible to be equal) because the i is 4 9 which is \ of 18 its cube 64 81 its square And since 81 is more than 64, the equation is im- possible and inept. " A few times Girard uses parentheses also to indicate multiplica- tion (see op. tit., folios Cf, D?, F. GERMAN-SPANISH: MARCO AUREL (1552) 165. Aurel states that his book is the first algebra published in Spain. He was a German, as appears from the title-page: Libra primer o de Arithmetica Algebratica ... por Marco Aurel, natural Aleman (Valencia, 1552). l It is due to his German training that German alge- braic symbols appear in this text published in Spain. There is hardly a trace in it of Italian symbolism. As seen in Figure 70, the plus (+) and minus ( ) signs are used, also the German symbols for powers of the unknown, and the clumsy Rudolffian symbols for roots of different 1 Aurel's algebra is briefly described by Julio Rey Pastor, Los malhemdlicos espaftoksdelsigloXVI (Oviedo, 1913), p. 36 n.; see Bibliotheca mathematica, Vol. IV (2d ser., 1890), p. 34. INDIVIDUAL WRITERS 161 orders. In place of the dot, used by Rudolff and Stifel, to express the root of a polynomial, Aurel employs the letter v, signifying universal root or rayz vniuersal. This v is found in Italian texts. Fio. 70. From Aurel's Arithmctica algebratica (1552). (Courtesy of the Li- brary of the University of Michigan.) Above is part of fol. 43, showing the -f and , and the radical signs of Rudolff, also the y'v. Below is a part of fol. 73B, con- taining the German signs for the powers of the unknown and the sign for a given number. 162 A HISTORY OF MATHEMATICAL NOTATIONS PORTUGUESE-SPANISH: PEDRO NUNEZ (1567) 166. Nunez' Libra de algebra (1567) 1 bears in the Dedication the date December 1, 1564. The manuscript was first prepared in the Portuguese language some thirty years previous to Nunez' prepara- tion of this Spanish translation. The author draws entirely from Italian authors. He mentions Pacioli, Tartaglia, and Cardan. The notation used by Nunez is that of Pacioli and Tartaglia. He uses the terms Numero, cosa, censo, cubo, censo de censo, relato primo, censo de cubo or cubo de censo, relato segundo, censo de censo de ceso, cubo de cubo, censo de relato primo, and their respective abbreviations co., ce., cu., ce.ce., re.p, ce.cu. or cu.ce., re.seg . ce.ce.ce., cu.cu., ce.re.p . He uses p for mas ("more"), and m for menos ("les"). The only use made of the %* is in cross-multiplication, as shown in the following , /f i At\ tt A.- i 12. 2.cu.p.8. sentence (fol. 41): ... partiremos luego y por y como si jL.cOm ji.ce, fuessen puros quebrados, multiplicado en *|, y verna por quociente 12 ce rt ' g el qual quebrado abreuiado por numero y por dignidad verna a este quebrado T~~"''>'J~'" This expression, multiplicando en *f, occurs often. Square root is indicated by R., cube root by R.cu., fourth root by R.R., eighth root by R.R.R. (fol. 207). Following Cardan, Nunez uses L.R. and R.V. to indicate, respectively, the ligatura ("combina- tion") of roots and the Raiz vniuersal ("universal root," i.e., root of a binomial or polynomial). This is explained in the following passage (fol. 456): "... diziendo assi: L.R.7pRA.p.3. que significa vna quanti- dad sorda compuesta de .3. y 2. que son 5. con la R.7. o diziendo assi: L.R.3p2.co. Raiz vniuersal es raiz de raiz ligada con numero o con otra raiz o dignidad. Como si dixessemos assi: R.v. 22 p 7? 9 ." Singular notations are 2. co. \. for 2[z (fol. 32), and 2. co. | for 2fz (fol. 366). Observe also that integers occurring in the running text are usually placed between dots, in the same way as was custom- ary in manuscripts. Although at this time our exponential notation was not yet in- vented and adopted, the notion of exponents of powers was quite well understood, as well as the addition of exponents to form the product 1 Libra de Algebra en arithmetica y Geometria. Compuesto por el Doctor Pedro Nunez, Cosmographo Mayor del Rey de Portugal, y Cathedratico Jubilado en la Cathedra de Mathematicas en la VniuerMad de Coymbra (En Anvers, 1567). INDIVIDUAL WRITERS 163 of terms having the same base. To show this we quote from Nunez the following (fol. 266) : "... si queremos multiplicar .4. co. por .5. ce. dircrnos asi .4. por .5. hazen .20. y porque .1. denominacio de co. sumado con .2. de- nominacion de censo hazen .3. quc cs denominacio de cubo. Diremos por tanto q .4. co. por .5. ce. hazen .20. cu. ... si multiplicamos .4. cu. por .8. ce.ce. diremos assi, la denominacion del cubo es .3. y la denominaciS del censo de censo es .4. q sumadas haze .7. q sera la denominacio dela dignidad engedrada, y por que .4. por .8. hazen .32. diremos por tanto, que .4. cu. multiplicados por .8. ce.ce. hazen .32. dignida-dcs, que tienen .7. por denominacion, a quc Hainan relatos segundos." Nunez' division 1 of 12z 3 +18x 2 +27a;+17 by 4z+3, yielding the I 1 2 quotient 3x 2 +2Jx+5 1 V+^ VQ > * s as ~- "Partidor A.co.p.3 I I2.cu.p.l8.ce.p.27.co.p.l7. 12.cu.p. 9.ce. 9. ce. p. 27. co. p. 17. O.ce.p. G.co.J. 20.co.-Jr.j5. 15 par A.co.p.3." Observe the "20.co.y for 20|x, the symbol for the unknown appearing between the integer and the fraction. _______ ___ Cardan's solution of x 3 +3x = 36 is ^1/325+ 18 -^1/325 -18, and is written by Nunez as follows: R. V.cu.R.325.p.l8.m.R. V .cu. .fl.325. w. 18. As in many other writers the V signifies vniversal and denotes, not the cube root of 1/325 alone, but of the binomial 1/325+18; in other words, the V takes the place of a parenthesis. 1 See II. Bosnians, "Sur le 'Libro de algebra' de Pedro Nunez," Bibliotheca mathematica, Vol. VIII (3d ser., 1908), p. 160-62; see also Tropfke, op. tit. (2d ed.), Vol. Ill, p. 136, 137. 164 A HISTORY OF MATHEMATICAL NOTATIONS ENGLISH: ROBERT RECORDE (i ^/i^r?! 1 f^7\ \lcrO[IJ, JuJOl ) 167. Robert Recorders arithmetic, the Grovnd of Aries, appeared in many editions. We indicate Recorders singular notation for pro- portion: 1 (direct) 3:8 = 16s.: 42s. 8d. 2s. 8d. Z 1 (reverse) There is nothing in Recorders notation to distinguish between the "rule of proportion direct" and the "rule of porportion reverse/' The difference appears in the interpretation. In the foregoing "direct" proportion, you multiply 8 and 16, and divide the product by 3. In the "reverse" proportion, the processes of multiplication and division are interchanged. In the former case we have 8X16-^3 = x, in the second case we have i-X-rV^iV^^- I* 1 both cases the large strokes in 2 serve as guides to the proper sequence of the numbers. 168. In Recorders algebra, The Whetstone of Witte (London, 1557), the most original and historically important is the sign of equality ( = ), shown in Figure 71. Notice also the plus (+) and minus ( ) signs which make here their first appearance in an English book. In the designation of powers Recorde uses the symbols of Stifel and gives a table of powers occupying a page and ending with the eightieth power. The seventh power is denoted by 6j"$; for the eleventh, thirteenth, seventeenth powers, he writes in place of the letter b the letters c, d, E, respectively. The eightieth power is de- noted by SJjgJs, showing that the Hindu multiplicative method of combining the symbols was followed. Figure 72 shows addition of fractions. The fractions to be added are separated by the word "to." Horizontal lines are drawn above and below the two fractions; above the upper line is written the new numerator and below the lower line is written the new denominator. In "Another Example of Addition," there are added the fractions 5s 6 +3s 5 , 20s 3 -6s 5 ~ r and yr~7i " Qx 9 bx* I 0p. tit. (London, 1646), p. 175, 315. There was an edition in 1543 which was probably the first. INDIVIDUAL WRITERS 165 Square root Recorde indicates by j/. or j/$, cube root by V\A/. or /vw/.cC- Following Rudolff, he indicates the fourth root by Thejfrte aa tfictc foojfef s Doe crtentte ) to &f ftinetc it ottclp fnf o ttooo parted Cfflftereof the firfte is, ifer <w* nombcris cqutlle wto one other. <3tlD tfte fccotlDc (0 >>6oi ow neat; irr is compared as ejtullc nt9,wtbcrn9mbers, fllluaics toillgngpou to rcmrbcr, tljatpou reduce sournombew, totficirleaftc Denominations, aim fmalltfte fo;mc^bcfo^e pott pjoce&c anp farther* flno again,if pour <!*** ion be focftc, tftat tfte grea^ tttte ucnomtnation G/?^ be toincD to anp partc of 4 compounDc nombcr 9 ?ou ftall tottrne It To , tfjat the tiombcroftbcgrcatcttc Co^ne alone, maicffanoca^ cquallctotficrcfte* ilnD tbts ts all tbat ncaoctb to be taugljtc , conccr- ,fo^ caffc altcratto of cftMtiom.% tuill p?0' pounoc a fctoc crapl5,b(caufe tf>c extraction of tfjetr rootc0,matc tt>c mo?c aptlp bee iu;ottgl)te flnD to a* uoiDctbctctrioufe repetition of tftcfe U)00^cs:t3c- qualle to : 3 tuill fette asj uoe often in tooo&efcfe,* paire of parallels, o: <Dcmotoc lines of one lengtbe, tl)us:^-===,btcaufe noe.2, cquallc. *I J.f -7 I* 20,t. - 26.5* I 6. $45. -- 12^40^ I 480^ 9.5- jn tbc firfte there appeared 2 nombcra , that ts I4.^/ FIG. 71. From Robert Recorded Whetstone of Witte (1557) -J but Recorde writes it also /vv/^- Instructive is the dialogue on these signs, carried on between master and scholar: 166 A HISTORY OF MATHEMATICAL NOTATIONS "Scholar: It were againste reason, to take reason for those signes, whiche be set voluntarily to signifie any thyng; although some tymes there bee a certaine apte conformitic in sochc thyngcs. And in these Jti other Example \)f Addition. fertcrme0* noe multiplication, no? reouttfon to one common DenomtnatojtftI) tbef bee one all reatp:no< tber ran the nombers be rcDuceD, to an? otber Icffcr^ but tbe quantities onel? be re&ticeD as pou fee* Scholar. 3|p;atet?ouletmep;oue jfn otberExamflf. \ 9 oaarttc pour iuo;fec tuell, before pou re* ourett. ^cbolar, 31 Teem? faulted ftauc frttc.2. nombcra fcucrallp, luttb one figne G/5' 4 ^; : bp reafon 31 DID not fojcf(c,tl)at,ct.multipJicD U)tti;,<.0oct^ maUe the tiUc FIG. 72. Fractions in Recorders Whetstone of Witte (1557) figures, the nomber of their minomes, seameth disagreable to their order. "Master: In that there is some reason to bee thewed: for as .j/. declareth the multiplication of a nomber, ones by it self; so ./wV- representeth that multiplication Cubike, in whiche the roote is repre- INDIVIDUAL WRITERS 167 sen ted thrise. And ./vsA standeth for .j/./\/. that is .2. figures of Square multiplication: and is not expressed with .4. minomes. For so should it seme to expresse moare then .2. Square multiplications. But voluntarie signes, it is inoughe to knowe that this thei doe signifie. f U F ; iW/f< :.. J*' : '* FJG. 73. Radicals in Recorded Whetstone of Witte (1557) And if any manne can diuise other, moare easie or apter in use, that maie well be received." Figure 73 shows the multiplication of radicals. The first two exercises are fOlXI^ 12 = 1^092, ^7|x^| = ^&t. Under fourth roots one finds V 15X^7== I^IOS . 168 A HISTORY OF MATHEMATICAL NOTATIONS ENGLISH: JOHN DEE (1570) 169. John Dee wrote a Preface to Henry Billingsley's edition of Euclid (London, 1570). This Preface is a discussion of the mathe- matical sciences. The radical symbols shown in Figure 74 are those of Stifel. German influences predominated. I^<5Hfet)^ nfibers, ^yS^pre & Ldfcas tlius </ft ii '+ Vc. if.Or thusV^S* i$> . &c.Arid&me tytne ivfth'ivhble numbers, or faftions of whole FIG. 74. Radicals, John Dee's Preface to Billingsley's edition of Euclid (1570). In Figure 75 Dee explains that if a:b = c:d, then also a:a b c:c d. He illustrates this numerically by taking 9:6 = 12:8. Notice Dee's use of the word "proportion" in the sense of "ratio." Attention is drawn to the mode of writing the two proportions 9.0:12.8 and 9.3:12.4, near the margin. Except for the use of a single colon (:), [ conucrfion ofproj^ttio^andoffomccucrfipnofproporrionJLA , berSjasp.to^fow.toS.c^ the cacccfTc of ^.thc antecedent of the firft proportion abouc 9.6: n . 8 ^thcconfcqaedtofdieramcis3: the txcdftpfii; the ante- o , a ; w 4, ccdcnt ofthcfecond proportion abouc 8,thec6ftfeqicnt of J ' * * xhcfime^4:nowc6pai^thcameccdeiuoftheftilpro nonp.as^tccedcttoj.thcexcciretherofaboue^v&cconfcqu^ ; fequent , Kkcwife compare i. thcahteccdcntoftheftcond proportion as antece- j detit to ^thcceffetherofalx)uc 8, the confequcnt^ to his con^ | yournumbersbein oysoidcr by conuerfion of proportion: as p,to 3 :fo 11^64: FIG. 75. Proportion in John Dee's Preface to Billingsley's edition of Euclid (1570). in place of the double colon ( : : ) , this is exactly the notation later used by Oughtred in his Clavis mathematicae. It is possible that Oughtred took the symbols from Dee. Dee's Preface also indicates the origin of these symbols. They are simply the rhetorical marks used in the text. See more particularly the second to the last line, "as 9. to 3: so 12. to 4:" INDIVIDUAL WRITERS 169 ENGLISH: LEONARD AND THOMAS DIGGES (1579) 170. The Stratioticos 1 was brought out by Thomas Digges, the son of Leonard Digges. It seems that the original draft of the book was the work of Leonard; the enlargement of the manuscript and its preparation for print were due to Thomas. The notation employed for powers is indicated by the following quotations (p. 33) : "In this Arte of Numbers Cossical, wae proceede from the Roote by Multiplication, to create all Squares, Cubes, Zenzizenzike, and Stir Solides, wyth all other that in this Science are used, the whyche by Example maye best bee explaned. 12345678 9 10 11 12 Roo. Sq. Cu. SqS. Sfo. SqC. Bfs. SSSq. CC. Sfs. CfS. 8SC. " 2 4 8 10 32 04 128 256 512 1024 2048 4090 Again (p. 32) : ". ... Of these [Roote, Square, Cube] are all the rest com- posed. For the Square being four, againe squared, maketh his Squared square 16, with his Character oner him. The nexte being not made by the Square or Cubike, Multiplication of any of the former, can not take his name from Square or Cube, and is therefore called a Surd solide, and is onely created by Multiplicand of 2 the Roote, in 16 the SqS. making 32 with his couenient Character ouer him & for distinctio is tearmed y first Surd solide .... the nexte being 128, is not made of square or Cubique Multiplication of any, but only by the Multiplication of the Squared Cube in his Roote, and therefore is tearmed the B.S.solide, or seconde S. solide 'This I have rather added for custome sake, bycause in all parts of the world these Characters and names of Sq. and Cu. etc. are used, but bycause I find another kinde of Character by my Father demised, farre more readie in Multiplications, Diuisions, and other Cossical operations, I will not doubt, hauing Reason on my side, to dissent from common custome in this poynt, and vse these Characters en- suing: [What follows is on page 35 and is reproduced here in Fig. 76]." 1 An Arithmeticall Militare Treatise, named Stratioticos: compendiously teaching the Science of Nubers, as well in Fractions as Integers, and so much of the Rules and A equations Algebraicall and Arte of Numbers Cossicall, as are requisite for the Profes- sion of a Soldiour. Together ivith the Moderne Militare Discipline, Offices, Lawes and Dueties in euery wel gouerned Campe and Armie to be observed: Long since attepted by Leonard Digges Gentleman, Augmented, digested, and lately finished, by Thomas Digges, his Sonne .... (At London, 1579). 170 A HISTORY OF MATHEMATICAL NOTATIONS FIG. 76. Leonard and Thomas Digges, Stratioticos (1579), p. 35, showing the unknown and its powers to x 9 . INDIVIDUAL WRITERS 171 As stated by the authors, the symbols are simply the numerals somewhat disfigured and crossed out by an extra stroke, to prevent confusion with the ordinary figures. The example at the bottom of page 35 is the addition of 20x+3Qx*+25x? and 45z+16o; 2 +13r'. It is noteworthy that in 1610 Cataldi in Italy devised a similar scheme for representing the powers of an unknown ( 340). The treatment of equations is shown on page 46, which is re- produced in Figure 77. Observe the symbol for zero in lines 4 and 7; this form is used only when the zero stands by itself. A little later, on page 51, the authors, without explanation, begin to use a sign of equality. Previously the state of equality had been expressed in words, "equall to," "are." The sign of equality looks as if it were made up of two letters C in these positions OC and crossed by two horizontal lines. See Figure 78. This sign of equality is more elaborate than that previously de- vised by Robert Recorde. The Digges sign requires four strokes of the pen; the Recorde sign demands only two, yet is perfectly clear. The Digges symbol appears again on five or more later pages of the Stratioticos. Perhaps the sign is the astronomical symbol for Pisces ("the Fishes"), with an extra horizontal line. The top equation on page 51 isz 2 ENGLISH: THOMAS MASTERSON (1592) 171. The domination of German symbols over English authors of the sixteenth century is shown further by the Arithmdicke of Thomas Masterson (London, 1592). Stifel's symbols for powers are used. We reproduce (in Fig. 79) a page showing the symbols for radicals. FRENCH: JACQUES PELETIER (1554) 172. Jacques Peletier du Mans resided in Paris, Bordeaux, Be- ziers, Lyon, and Rome. He died in Paris. His algebra, De occvlta Parte Nvmerorvm, Quam Algebram vocant, Libri duo (Paris, 1554, and several other editions), 1 shows in the symbolism used both German and Italian influences: German in the designation of powers and roots, done in the manner of Stifel; Italian in the use of p. and m. for "plus" and "minus." 1 All our information is drawn from H. Bosnians, "L'algebre de Jacques Peletier du Mans," Extrait de la revue des questions scientifiques (Bruxelles: Janu- ary, 1907), p. 1-61. 172 A HISTORY OF MATHEMATICAL NOTATIONS on) to $> oa t&ewiNt#, t&ere maotiw fome ffl ttn&rr (a -^omctimf* Rcdmfbon fct maUebpuwjina foitfth?r all ' wwjis^^ '"^J' " ^4*' ~ ' J ti" ~ **** "Or % l ' i*r^rti'*ti4i^i&]^ A. HL+ I'.^K'JUtttra.'S&i^ iiM':''*-*i ati.i^iii, ^.'liJr f! *Tr**Tf 4 t.' ^gii*ii Fia. 77. Equations in Digges, Stratioticos (1579) INDIVIDUAL WRITERS 173 J FIG. 78. Sign of equality in Digges, Straiioticos (1579). This page exhibits also the solution of quadratic equations. AMTHMBTICKB. LIB. 75 "* 80 9 f ^50 /4f/f i 4 48 27 8 1 + oi FIG. 79. Thomas Masterson, Arithmeticke (1592), part of p. 45 174 A HISTORY OF MATHEMATICAL NOTATIONS Page 8 (reproduced in Fig. 80) is in translation: "[The arith- metical progression, according to the natural order of counting,] furnishes us successive terms for showing the Radicand numbers and their signs, as you see from the table given here [here appears the table given in Fig. 80]. : ''T';^ V r ^U'^v 1 ; 1 , ~'i ~ ', ' ,.,/-,*- . _ . , - i - ' , j < ' ^ M "-i!"*;s.'V ' '- * " ; ../' ;U < r " " ' ; M ' /^yl*.V-:"'-' J .'t ;', ; OiiJr */V'4 1* :? -7 .-.I, K, |, I/I, ,* .-; ,f, iy 4, S, 'itf, lit, jfcjtf, f^ ', : j>i i'^t,*-' /ij, ^14 if,' V^c^ ' T . ,; .,: ; - l ' : ^' :l ; 1 :' > ,r-" - ' / , " ; v,^:-, M ' ; ^ , ! $t & ! | i ' E qrii ft 5H i;fe| 4c IP ^ r |f ijy ^ ; " 4c c^ 1^ ; : ; ( ! ." ' ;. ; V I! FIG. 80, Designation of powers in J. Peletier's Algebra (1554) "In the first line is the arithmetical progression, according to the natural order of the numbers; and the one which is above the & numbers the exponent of this sign & ; the 2 which is above the 3 is the exponent of this sign 3 ; and 3 is the exponent of c, 4 of 33, and so on. "In the second line are the characters of the Radicand numbers INDIVIDUAL WRITERS 175 which pertain to algebra, marking their denomination." Then are ex- plained the names of the symbols, as given in French, viz., 1} ratine, 3 Qanse, < cube, etc. FIG. 81. Algebraic operations in Peletier's Alytbra (1554) Page 33 (shown in Fig. 81) begins with the extraction of a square root and a "proof" of the correctness of the work. The root extraction is, in modern symbols: 36z 4 +48z 3 - 104x 2 - 80x+ 100 + 12x 2 + 8z- +120z 2 +80z-100. 176 A HISTORY OF MATHEMATICAL NOTATIONS The "proof" is thus: 4x~ 10 6z 2 + 4x - 10 36z 4 +24z 3 - 60s 2 16s 2 -' - 60z 2 -40z+100 Further on in this book Peletier gives: 1/3 15 p. |/j8, signifying 1/15+1/8 . l/j . 15 p. i/j8, signifying ^15+1/8 . FRENCH: JEAN BXJTEON (1559) 173. Deeply influenced by geometrical considerations was Jean Buteon, 1 in his Logistica quae et Arithmetica vulgo dititur (Lugduni, 1559). In the part of the book on algebra he rejects the words res, census, etc., and introduces in their place the Latin words for "line," "square," "cube," using the symbols p, <>, Q). He employs also P and M y both as signs of operation and of quality. Calling the sides of an equation continens and contentum, respectively, he writes between them the sign [ as long as the equation is not reduced to the simplest form and the contentum t therefore, not in its final form. Later the contentum is inclosed in the completed rectangle [ ]. Thus Buteon writes 3p M 7 [ 8 and then draws the inferences, 3p [15], lp [5]. Again he writes | <> [100, hence 1<> [400], lp [20]. In modern symbols: 3z-7 = 8, 3z = 15, re = 5; iz 2 <=100, ^ = 400, x = 20. Another example: 1 Q P 2 [218, i a [216, 1 O [1728], lp [12]; in modern form la; 3 +2 = 218, i* 3 = 216, z* = 1,728, x = 12. When more than one unknown quantity arises, they are repre- sented by the capitals A, B y C. Buteon gives examples involving only positive terms and then omits the P. In finding three numbers sub- ject to the conditions x+^y+^z** 17, y+&+ 1*= 17, z+-x+y = 17, he writes: IA , \B , |C [17 IB , tA , iC [17 1C , \A , \B [17 1 Our information is drawn from G. Werthheim's article on Buteon, Biblio- theca mathematica, Vol. II (3d ser., 1901), p. 213-19. INDIVIDUAL WRITERS and derives from them the next equations in the solution: 2A . IB . 1C [34 1A . 35 . 1C [51 1A . IB . 4C [68, etc. 177 $m."aM*\i^i4ffi7* ***_ "it 5 ,fitirfjidito 4,f) FIG. 82. From J. Buteon, Arilhmetica (1559) 178 A HISTORY OF MATHEMATICAL NOTATIONS In Figure 82 the equations are as follows: 34 + 125+ 3(7= 96 3A+ 1B+ 1C- 42 2(7= 54 3A+ 35+ 15(7 = 120 3A+ 15+ 1(7 = 42 25+ 14C = 78 225+154(7 = 858 225+ 4(7=108 150(7 = 750 FRENCH: GUILLAUME GOSSELIN (1577) 174. A brief but very good elementary exposition of algebra was given by G. Gosselin in his De arle magna, published in Paris in 1577. Although the plus (+) and minus ( ) signs must have been more or less familiar to Frenchmen through the Algebra of Scheubel, published in Paris in 1551 and 1552, nevertheless Gosselin does not use them. Like Peletier, Gosselin follows the Italians on this point, only Gosselin uses the capital letters P and M for "plus" and "minus," instead of the usual and more convenient small letters. 1 He defines his notation for powers by the following statement (chap, vi, fol. v) : L - 2 - Q 4 C - 8 QQ 16 RP 32 QC 64 - RS 128 - CC 512 . Here RP and RS signify, respectively, relatum primum and relatum secundum. Accordingly, 11 12L M IQ P 48 aequalia 144 M 24L P 2Q " means *Our information is drawn mainly from H. Bosnians' article on Gosselin, Bibliotheca mathematica, Vol. VII (190(>-7), p. 44-66. INDIVIDUAL WRITERS 179 The translation of Figure 83 is as follows: " . . . . Thus I multiply 4z-6z 2 +7 by 3x 2 and there results 12Z 3 18z 4 +21a; 2 which I write below the straight line; then I multi- i }',' H' 'i' '' ' 1 ' , Ij ti-w^i|> r ' jT if ^ i f Mt IK Ji r* f '.iff f ^Sr " sM) "'"" T f 1 1;, ITV ^~i jti ^>'y& j$.ihkxGt&m&' - FIG. 83. Fol. 45t> of Gosselin's Z>e arie magna (1577) ply the same 4z 6o; 2 +7 by +4x, and there results +16x 2 24^ +28z; lastly I multiply by 5 and there results 20x+30x 2 35. 180 A HISTORY OF MATHEMATICAL NOTATIONS And the sum of these three products is 67x*+8x 12s 3 18s 4 35, as will be seen in the example. 4z - 6z 2 + 7 3z 2 + 4x - 5 ( 12z 3 -18o; 4 +21z 2 Products \ 16z 2 - 24z 3 +28z [-20s+30z 2 -35 Sum 67z 2 +8z-12z 3 -18z 4 -35 . On the Division of Integers, chapter viii Four Rules + divided in + the quotient is + divided in the quotient is + divided in + the quotient is + divided in the quotient is " 175. Proceeding to radicals we quote (fol. 475): "Est autem laterum duplex genus simplicium et compositorum. Simplicia sunt 19, LC8, LL16, etc. Composita vero ut LF24 P L29, LF6 P L8." In translation: "There are moreover two kinds of radicals, simple and composite. The simple are like 1/9, f^8, 1/16, etc. The com- posite are like */24+)/29, ^G+l/S." First to be noticed is the dif- ference between L9 and 9L. They mean, respectively, 1/9 and Qx. We have encountered somewhat similar conventions in Pacioli, with whom & meant a power when used in the form, say, "5 . 5?" (i.e., x 4 ), while B, meant a root when followed by a number, as in # .200. (i.e., 1/200) (see 135). Somewhat later the same principle of relative position occurs in Albert Girard, but with a different symbol, the circle. Gosselin's LV meant of course latus universale. Other examples of his notation of radicals are L7L10 P L5, for */VlO+ 1/5, and LFCL5 PLC10 for In the solution of simultaneous equations involving only positive terms Gosselin uses as the unknowns the capital letters A, B, C, . . . . (similar to the notation of Stifel and Buteon), and omits the sign P for "plus"; he does this in five problems involving positive terms, following here an idea of Buteo. In the problem 5, taken from Buteo, Gosselin finds four numbers, of which the first, together with half of the remaining, gives 17; the second with the third of the remaining gives 12; and the third with a fourth of the remaining gives 13; and INDIVIDUAL WRITERS 181 the fourth with a sixth of the remaining gives 13. Gosselin lets A, B t C, D be the four numbers and then writes: Modern Notation lia 17 , x+^y+{z+^w = l7 , lia 12, etc. " y+fc+fc+b = 12 . He is able to effect the solution without introducing negative terms. In another place Gosselin follows Italian and German writers in representing a second unknown quantity by q, the contraction of quantitas. He writes (fols. 84#, 85A) "1L P 2q M 20 aequalia sunt 1L P 30" (i.e., lz+22/-20=lz+30) and obtains "2q aequales 50, fit 1^25" (i.e., 2y = 50, i/ = 25). FRENCH: FRANCIS VIETA (1591 and Later) 176. Sometimes, Vieta's notation as it appears in his early publi- cations is somewhat different from that in his collected works, edited , a , , . __ , , 3Z) 2 -3#A 2 by Fr. van Schooten in 1646. For example, our modern - ^ - is printed in Vieta's Zeteticorum libri v (Tours, 1593) as " B in D quadratum 3 B in A quadratum 3 " _ f while in 1646 it is reprinted 1 in the form "gin DqZ-Bm A? 3 >f 4 Further differences in notation are pointed out by J. Tropfke: 2 Zeteticorum libri v (1593) f B in A 1 ^ . OT> " B in A , I -B in H I , , D Fol. 3B: TT -- 1- s - pj - > aequabuntur B ." V I * ) , T , Bx , Bx-B H D Modern : -_- -\ -- ^ - = B . LJ r OK K Lib. II, 22: i_i?. o o 1 Francisci Victae Opera mathemalica (ed. Fr. & Schooten; Lvgdvni Batavorvm, 1646), p. 60. This difference in notation has been pointed out by H. Bosnians, in an article on Oughtred, in Extrait des annales de la soci&& scientifique de Bruxelles, Vol. XXXV, fasc. 1 (2d part), p. 22. 2 Op. cit., Vol. Ill (2d ed., 1922), p. 139. 182 A HISTORY OP MATHEMATICAL NOTATIONS Modern: B(D*+BD) . Lib. IV, 20: - D in (^cubum2l ' i D cubo J Modern: D(2B 3 -D*) . Van Schooten edition of Vieta (1646) ^ . " B in A , B in A-B in # ... D P. 46: ^ -- 1 -- ET - aequabitur B ." D r /25 /5 \y \3- P. 70: " B in D quad.+# in D ." P. 74: " D in 5 cubum 2-D cubo ." Figure 84 exhibits defective typographical work. As in StifePs Arithmetica Integra, so here, the fractional line is drawn too short. In the translation of this passage we put the sign of multiplication (X) in place of the word in: ". . . . Because what multiplica- tion brings about above, the same is undone by division, as ^ . i.e.. jj. \ and ^ is A. * o r> ... A 2 A 2 -\-ZXB Thus in additions, required, to -~ to add Z. The sum is - ^ - ; . . A 1 . , , Z 2 . or required, to -^ to add 77 . The sum is > Or A 2 In subtraction, required, from -^- to subtract Z. The remainder is n . . . A 2 , u , . Z 2 _ . , . Q r required, from -^ to subtract -~ . Ine remainder is > Cr Observe that Vieta uses the signs plus (+) and minus (), which had appeared at Paris in the Algebra of Scheubel (1551). Outstanding in the foregoing illustrations from Vieta is the appearance of capital letters as the representatives of general magnitudes. Vieta was the first to do this systematically. Sometimes, Regiomontamus, Rudolff, Adam Riese, and Stifel in Germany, and Cardan in Italy, used letters at an earlier date, but Vieta extended this idea and first made it an INDIVIDUAL WRITERS A 183 essential part of algebra. Vieta's words, 1 as found in his Isagoge, are: "That this work may be aided by a certain artifice, given magnitudes are to be distinguished from the uncertain required ones by a symbol- ism, uniform and always readily seen, as is possible by designating the required quantities by letter A or by other vowel letters A y l y y V, Y, and the given ones by the letters #, (?, D or by other consonants," 2 Vieta's use of letters representing known magnitudes as coeffi- cients of letters representing unknown magnitudes is altogether new. In discussing Vieta's designation of unknown quantities by vowels, [Tluper cttecit multiplicatio , idem rcioiuit diuilid vf'B W~ A~ 7 14i 4 % "ff : g W I eft A planum , , > ' ~H ' V '""IP* ' ** ! < t t ? - ' v> * > **(? lJ ' l&l' t7 1 * ' ' ' x Woe in AddfcioBibw* , Oportei* A jftyw fdifcr* Vj^^w jafc? ^ffiW- ' <!' f I i " JV * A plano-i^duccrc ,^ qm4rtQiro. ; Rcfiduaerit t Aipittt,ite"G ^ t!* t~~~ =m^^ ^ t .. * <> x^' ,'.>, M ,. Jk .j -A ^ i.. t> } ,* FIG. 84. From Vieta's In arlem analyticam Isagoge (1591). (I am indebted to Professor H. Bosnians for this photograph.) C. Henry remarks: "Thus in a century which numbers fewer Oriental- ists of eminence than the century of Viet a, it may be difficult not to regard this choice as an indication of a renaissance of Semitic lan- guages; every one knows that in Hebrew and in Arabic only the conso- nants are given and that the vowels must be recovered from them/' 3 177. Vieta uses = for the expression of arithmetical difference. He says: "However when it is not stated which magnitude is the greater and which is the less, yet the subtraction must be carried out, 1 Vieta, Opera mathematica (1646), p. 8. 2 "Quod opus, ut arte aliqua juvetur, symbolo constant! et perpetuo ac bene 6onspicuo date magnitudines ab incertis quaesititiis distinguantur, ut pote magni- tudines quaesititias elemento A aliave litera vocali, E, 7, O, V, Y datas elementis Bj G, D t altisve consonis designando." 3 "Sur Forigine de quelques notations mathe'matiques," Revue archfologique, Vol. XXXVIII (N.S., 1879), p. 8. 184 A HISTORY OF MATHEMATICAL NOTATIONS the sign of difference is =, i.e., an uncertain minus. Thus, given A 2 and B 2 , the difference is A 2 B 2 , or 2 =AV' 1 We illustrate Vieta's mode of writing equations in his Isagoge: "B in A quadratum plus D piano in A aequari Z solido," i.e., BA 2 + D 2 A = Z 3 , where A is the unknown quantity and the consonants are the known magnitudes. In Vieta's Ad Logisticen speciosam notae priores one finds: "A cubus, +A quadrato in B ter, +A in B quadratum ter, +B cubo," for A*+3A 2 B+3AB 2 +B 32 We copy from Vieta's De emendatione aequationum tradatus secun- dus (1615) , 3 as printed in 1646, the solution of the cubic x*+W*x = 2Z 3 : "Proponatur A cubus + B piano 3 in A, aequari Z solido 2. Oportet facere quod propositum est. E quad. +A in E, aequetur B piano. Vnde B planum ex hujus modi aequationis const itutione, in- telligitur rectangulum sub duobus lateribus quorum minus est E t ,.,- ,. v . A . ., B planum E quad. ., . ~ differentia & majore A. igitur - - ^ -- - - erit A. Quare Hi B plano-plano-planum E quad, in B plano-planum 3+E quad. quad, in B planum 3 E cubo-cubo . 5 pi. pi. 3. B pi. in Eq. 3 - - -- 1 ^^ - E~ aequa- bitur Z solido 2 . "Et omnibus per E cubum ductis et ex arte concinnatis, E cubi quad.+Z solido 2 in E cubum, aequabitur B plani-cubo. 4 "Quae aequatio est quadrati affirmate affecti, radicem habentis solidam. Facta itaque reductio est quae imperabatur. "Confectarium: Itaque si A cubus + B piano 3 in A, aequetur Z solido 2, & VE plano-plano-plani + Z solido-solido Z solido, 7^ u T-. B planum D quad. ., A , ., aequetur D cubo. Ii.rgo I -- ^ - - - , sit A de qua quaeritur. Translation : "Given x*+3B 2 x = 2Z 3 . To solve this, let y*+yx = B 2 . Since B 2 from the constitution of such an equation is understood to be a rectangle of which the less of the two sides is y, and the difference B 2 y 2 between it and the larger side is x. Therefore ----- ~ x. Whence . y 1 y 1 "Cum autem non proponitur utra magnitude sit major vel minor, et tamen subductio facienda est, nota differentiae est ~ id est, minus incerto: ut propositis A quadrato et B piano, differentia erit A quadratum :zz B piano, vel B planum A zn quadrato" (Vieta, Opera mathematica [1646], p. 5). 2 Ibid., p. 17. 3 Ibid., p. 149. 4 " plani-cubo" should be "B cubo-cubo," and "E cubi quad." should be "E cubo-cubo." INDIVIDUAL WRITERS 185 All terms being multiplied by ?/ 8 , and properly ordered, one obtains 7/ 6 +2Z 3 i/ 3 = B G . As this equation is quadratic with a positive affected term, it has also a cube root. Thus the required reduction is effected. "Conclusion: If therefore x 3 +3J5 2 a; = 2Z 3 , and T/B+Z-Z* = D*, B 2 D 2 . then ,. is x, as required." The value of x in x?+3B 2 x = 2Z* is written on page 150 of the 1646 edition thus: plano-plano-plani+Z solido-solido+Z solido * C.VB plano-plano-plani+# solidoTSolido. Z solido ." The combining of vinculum and radical sign shown here indicates the influence of Descartes upon Van Schooten, the editor of Vieta's collected works. As regards Vieta's own notations, it is evident that compactness was riot secured by him to the same degree as by earlier writers. For powers he did not adopt either the Italian symbolism of Pacioli, Tartaglia, and Cardan or the German symbolism of lludolff and Stifel. It must be emphasized that the radical sign, as found in the 1646 edition of his works, is a modification introduced by Van Schooten. Vieta himself rejected the radical sign and used, instead, the letter I (latus, "the side of a square") or the word radix. The I had been introduced by Ramus ( 322) ; in the Zetetic&rum, etc., of 1593 Vieta wrote 1. 121 for 1/121. In the 1646 edition (p. 400) one finds V 2 +V 2 + l// 2+l/2, which is Van Schooten's revision of the text of Vieta; Vieta's own symbolism for this expression was, in 1593, l "Radix binomiae 2 (2 (2 + Radix binomiae \+ radix binomiae \+radice 2 ," and in 1595, 2 " R. bin. 2+R. bin. 2+R. bin. 2+R. 2. ," a notation employed also by his contemporary Adrian Van Roomen. 178. Vieta distinguished between number and magnitude even in his notation. In numerical equations the unknown number is no longer represented by a vowel ; the unknown number and its powers are repre- sented, respectively, by N (numerus), Q (quadratus), C (cubus), and 1 Variorum de rebus mathem. Responsorum liber VIII (Tours, 1593), corollary to Caput XVIII, p. 12v. This and the next reference are taken from Tropfke, op. ciL, Vol. II (1921), p. 152, 153. 2 Ad Problema quod omnibus mathematicis totius orbis conslruendum proposuit Adrianus Romanus, Francisci Vietae responsum (Paris, 1595), Bl. A IV. 186 A HISTORY OF MATHEMATICAL NOTATIONS combinations of them. Coefficients are now written to the left of the letters to which they belong. Thus, 1 "Si 65C-1QQ, aequetur 1,481,544, fit 1#57," i.e., if 65^-^=1,481,544, then z = 57. Again, 2 the "B3 in A quad." occur- ring in the regular text is displaced in the accompanying example by "6Q," where = 2. Figure 85 further illustrates the notation, as printed in 1646. Vieta died in 1603. The De emendations aeqvationvm was first printed in 1615 under the editorship of Vieta's English friend, Alex- ander Anderson, who found Vieta's manuscript incomplete and con- THBORBMA I, Si A cubtis H- B in A quadr. 3 -* D piano in A , sequetur B cubo z D piano in B. A quad. H- BmAi>xquabiturBquad.i D planer. Quoniamcnim A quadr. -h B in A i, aecjbatur B quadr. i D piano. Du&is igicut omnibus in A. A cubus -t- B in A quad, i, sequabicur B quad, in A z D piano in A . Et iifdcm du&is in B. B in A quad, -f B quadr. in A i , gquabuur B cubo i D piano in B. lungatur dufta xqualia xqualibus. A cubus + B in A quad. 3 H- B quad, in A 2 , acquabicur B quad.in A z D piano in A 4- B cubo i D piano in B. EtdclctaumnqueadfcftioneBquad.in A i, &adacqualitatisordinationcm, tranfla- raperamithcfin Oplaniin A adfedionc. A cubus -t- B in A quadr. 5 -t- D piano in A, acquabiturB cuboi D piano in B. Quodquidcmitafchabct. FIG. 85. From Vieta's De emendations aeqvationvm, in Opera mathematica (1646), p. 154. taining omissions which had to be supplied to make the tract intelli- gible. The question arises, Is the notation AT, Q, C due to Vieta or to Anderson? 3 There is no valid evidence against the view that Vieta did use them. These letters were used before Vieta by Xylander in his edition of Diophantus (1575) and in Van Schooten's edition 4 of the Ad problema, quod omnibus mathematicis totius orbis construendum proposuit Adrianus Romanus. It will be noticed that the letter N stands here for x, while in some other writers it is used in the designa- tion of absolute number as in Grammateus (1518), who writes our 12x 3 -24 thus: "12 ter. mi. 24/V." After Vieta N appears as a mark for absolute number in the Sommaire de Valgebre of Denis Henrion 5 1 Vieta, Opera mathematica (1646), p. 223. 2 Op. cit., p. 130. 3 See Enestrom, Bibliotheca mathematica, Vol. XIII (1912-13), p. 166, 167. 4 Vieta, Opera mathematica (1646), p. 306, 307. 6 Denis Henrion, Les qvinze livres des elemens d'Evclide (4th ed.; Paris, 1631), p. 675-788. First edition, Paris, 1615. (Courtesy of Library of University of Michigan.) INDIVIDUAL WRITERS 187 which was inserted in his French edition of Euclid. Henrion did not adopt Vieta's literal coefficients in equations and further showed his conservatism in having no sign of equality, in representing the powers of the unknown by ft, q, c, qq, ft, qc, bfi, qqq, cc, q/3, eft, qqc, etc., and in using the "scratch method" in division of algebraic polynomials, as found much earlier in Stifel. 1 The one novel feature in Henrion was his regular use of round parentheses to express aggregation. ITALIAN: BONAVENTURA CAVALIERI (1647) 179. Cavalieri's Geometria indivisibilibvs (Bologna, 1635 and 1653) is as rhetorical in its exposition as is the original text of Euclid's Elements. No use whatever is made of arithmetical or algebraic signs, not even of + and , or p and m. An invasion of German algebraic symbolism into Italy had taken place in Clavius' Algebra, which was printed at Rome in 1608. That German and French symbolism had gained ground at the time of Cavalieri appears from his Exercilationes geometriae sex (1647), from which Figure 86 is taken. Plus signs of fancy shape appear, also Vieta's in to indicate "times." The figure shows the expansion of (a+fr) n for n = 2, 3, 4. Observe that the numerical coefficients are written after the literal factors to which they belong. ENGLISH: WILLIAM OUGHTRED (1631, 1632, 1657) 180. William Oughtred placed unusual emphasis upon the use of mathematical symbols. His symbol for multiplication, his notation for proportion, and his sign for difference met with wide adoption in Continental Europe as well as Great Britain. He used as many as one hundred and fifty symbols, many of which were, of course, intro- duced by earlier writers. The most influential of his books was the Clavis mathematicae, the first edition 2 of which appeared in 1631, later Latin editions of which bear the dates of 1648, 1652, 1667, 1693. 1 M. Stifel, Arithmetica Integra (1544), fol. 239A. 2 The first edition did not contain Clavis mathematicae as the leading words in the title. The exact title of the 1631 edition was: Arithmeticae in\numeris et sped-\ ebvs institvtio:\Qvae tvm loyislicae, tvm analyli\cae, atqve adeo\(olivs mathematical', qvasi\davis\esL\ Ad nobilissimvm spe\ctatissimumque iuvenem I)n. Ovilel\mvm Howard, Ordinis, qui dici\tur, Balnei Equitem, honoratissimi Dn.\ Thomac, Comitis Arvndeliae & \ Svrriae, Comitis Mareschal\li Angliae, &c. filium. \Lon- dini,\Apud Thomam Harpervm,\ M. DC. xxxi. 188 A HISTORY OF MATHEMATICAL NOTATIONS A second impression of the 1693 or fifth edition appeared in 1698. Two English editions of this book came out in 1647 and 1694. * '''-. ,.a^^'-;"" FIG. 86. From B. Cavalieri's Exercitationes (1647), p. 268 We shall use the following abbreviations for the designation of tracts which were added to one or another of the different editions of the Clavis mathematicae: Eq. = De Aequationum affectarvm resolvtione in numeris Eu. = Ekmenti decimi Euclidis declaratio So. De Solidis regularibus, tradatus An. -De Anatotismo, sive usura composita Fa. = Regula falsae positionis Ar. = Theorematum in libris Archimedis de sphaera & cylindro declaratio Ho. = Horologia scioterica in piano, Geometries delineandi modus INDIVIDUAL WRITERS 189 In 1632 there appeared, in London, Oughtred's The Circles of Proportion, which was brought out again in 1633 with an Addition vnto the Vse of the Instrument called the Circles of Proportion. 1 Another edition bears the date 1660. In 1657 was published Oughtred's Trigonometria, 2 in Latin, and his Trigonometrie, an English transla- tion. We have arranged Oughtred's symbols, as found in his various works, in tabular form. 3 The texts referred to are placed at the head of the table, the symbols in the column at the extreme left. Each number in the table indicates the page of the text cited at the head of the column containing the symbol given on the left. Thus, the nota- tion : : in geometrical proportion occurs on page 7 of the Clavis of 1648. The page assigned is not always the first on which the symbol occurs in that volume. 1 In our tables this Addition is referred to as Ad. 2 In our tables Ca. stands for Comones sinuum tangentium, etc., which is the title for the tables in the Trigonometria. 3 These tables were first published, with notes, in the University of California Publications in Mathematics, Vol. I, No. 8 (1920), p. 171-80. 190 A HISTORY OF MATHEMATICAL NOTATIONS 181. OUGHTRED'S MATHEMATICAL SYMBOLS SYMBOLS MEANINGS OF SYMBOLS Clans mathematica. ti *< ^ ? Opue. Po*th. r 1677 | 1631 1647 1648 1652 1667 1693 1694 sS' S 0[56 0.56 .[58 0,56 OpOOOo a.6 2.314 2,314 2.314 .S + P mo mi e 1 X Equal to Separatrix' Separatrix Separatrix 3 Separatrix .00005 Ratio a: 6, or *a 6 ] [Separating 5 j | the mantissa Characteristic Arithm. proportion 8 0:6, ratio 7 Given ratio Geomet. proportion 8 Contin. proportion Contin. proportion Geom. 9 proportion ( )" ( ( ) ( ) ( ) ( )" ( )" ( ) Therefore Addition" Addition Addition 1 ' Subtraction Plus or minus Subtraction Less" Negative 2 Multiplication 15 38 1 34 1 53 1 30 15 1 16 1 73 2 20 3 3 13 235 3 63 29 1 17 5 2 27 ' 221 3 7 3 12 3 7 3 7 5 8 25 7 4 3 235 3 Eq.im &7.187 158 158 22 4n.l62 150 150 21 113 150 21 113 150 21 175 207 32 24 49 11 25 10 19 7 36 3 34 140 87 3 142 42 27 29 114 89 75 53 98 97 116 101 81 5 5 21 5 13 28 8 18 33 7 16 32 7 16 25 7 16 25 7 16 45 40 57 58 115 58 65 107 99 106 58 104 92 104 95 57 52 56 104 95 57 53 92 104 63 63 An. 42 149 119 95 122 95 97 96 35 32 101 101 102 89 151 3 112 2 49 3 3 3 3 57 57 3 3 3 3 4 4 4 4 140 4 4 16 13 99 96 3 2 51 3 57 3 3 106 66 10 56 57 3 53 3 3 17 3 21 96 3 16 4 130 5 97 7 1 10 9 10 1 10 10 5 10 8 37 32 143 INDIVIDUAL WRITERS 182. OUGHTRED'S MATHEMATICAL SYMBOLS Cont. 191 SYMBOLS MEANINGS OP SYMBOLS Claris rnathdndticcif a. co o * 5 s ^ 1 .^ II . t-- P 1 1631 1647 1648 1652 1667 1693 1694 Hqbq in I a)b(c flilS Aq Ac Aqq Aqc Ace ABq [*}.... m 14] .... [10] a 2 . . . . a? a Q (lu c Cu 0,0, oe D 1,1 L AL p p R I*,R R V V v VA X By juxtaposition Multiplication" Fraction, division 6-7-a=c I + J-t AA AAA vl^AA AAAAA A/UAAA ZB*" 4th .... 10th power 4th 10th power a* . . . . a' Quaesitum Square" Square Cube Cube 4th power 5th power Diameter Latw, radix 19 Angle Angles Perimeter ZA-Aq Radius Remainder Rational Superficies curva Root Square root Square root r xi<ufi binomii 7 1 8 10 11 10 12 14 10 10 11 13 11 10 11 14 10 10 11 14 37 10 11 13 13 23 21 96 21 5 16 87 219 9 99 156 104 105 106 17 59 5 50 17 25 41 67 41 7 7 7 7 7 23 11 11 11 11 11 11 37 10 10 10 10 10 11 35 10 10 10 10 10 11 34 35 10 10 10 10 10 11 34 35 10 10 10 10 10 11 14 14 14 55 55 15 53 65 34 52 205 24 38 17 33 16 31 16 57 16 30 16 30 25 47 28 5 100 75 105 53 38 45 33 136 33 33 31 128 61 31 187 30 123 30 30 Eu 21 30 123 30 30 Eu. 21 30 123 30 30 Eu 20 47 175 47 47 28 62 210 210 37 37 121 113 110 110 110 158 139 19 16 192 37 41 120 152 111 134 166 109 126 Eu. 1 AT I 109 128 Eu. 1 Ar 1 109 142 Eu. 1 Ar 1 154 37 32 211 45 35 33 53 49 33 31 48 48 31 30 47 46 30 30 47 46 30 30 47 46 30 47 70 65 47 102 134 96 192 A HISTORY OF MATHEMATICAL NOTATIONS 183. OUGHTRED'S MATHEMATICAL SYMBOLS Cont. SYMBOLS MEANINGS or SYMBOLS Claris mathematicae -* ^.^ f t 1 i i & *" QO 1631 1647 1648 1652 1667 1693 1694 Vr Vu VQ</ Vc V 9 c Vcc V ccc V cccc VQU V[121or1 vQ'j! r</, re r, ru A,E z X z X z 9C c --3 tr _b C r- rr Aft < > ~cu "no. Lotus residui Sq. rt. of polyno. 80 4th root Cube root 5th root 6th root 9th root 12th root Square root 12th root v, f Square root Nos., A>E A+" A-E A*+E> A>-E> A3+E> A*-E a-\-e ae a'+fc 1 a'-b' Majus** Minus Non majut Non minus Minus** Minus** Major ratio Minor ratio Less than* 1 Greater than Commenaurabilia Incommensurabilia 35 35 35 37 34 55 52 52 49 52 31 53 47 49 47 49 30 53 46 46 46 48 30 52 46 46 46 48 30 52 48 46 46 48 47 96 69 69 65 69 37 49 37 52 49 48 48 48 69 52 50 49 49 49 69 73 74,96 53 53 53 54 54 94 94 21 21 21 41 41 44 44 33 33 33 33 33 33 33 31 31 31 31 31 31 31 167 30 30 30 30 30 30 30 Eu. 1 Eu. 1 30 30 30 30 30 30 30 EuA EuA 30 30 30 30 30 30 30 EuA EuA 47 47 47 47 47 47 47 87 87 87 98 99 19 16 16 167 167 166 166 166 166 Eu.2 Eu.2 145 EuA Eu.l Eu.l Eu.2 Eu.2 EuA EuA EuA EuA EuA EuA EuA EuA EuA EuA Ho, 17 Ho. 17 I Ho. 30 166 166 Ho. 31 Eu. 1 Eu. 1 Ho. 29 EuA EuA EuA EuA 11 6 4 4 166 166 Eu.l Eu.l EuA EuA Eu.l EuA 1 INDIVIDUAL WRITERS 184. OUGHTRED'S MATHEMATICAL SYMBOLS Cont. 193 SYMBOLS MEANINGS or SYMBOLS Claris mathematicae "? - ^s" a i i fri*" i il S 1 i?"" 1 O 1631 1647 1648 1652 1667 1693 1694 * V V nT r <r T eim cr _o = SSJ o A 4 v^*\ 11 log log:^: S t se 8V t ver sin : com SCO too 86 CO sin tan sec sec:parall Comment, potentia Incommens. potentia Rationale Irrationale Medium Line, cut extr. and mean ratio Major ejus portio Minor ejus portio Simile Proxime majus Proxime minus Aequale vel minut Aeqitale vel ma jus Rectangulum Quadratum Trianguhan Latus, radix Media proportion Differentia^* Parallel Logarithm Log. of square Sine" Tangent Secant Sinus versut Sinus versus Sine complement Cosine Cotangent Cosecant Sine Tangent Secant Sum of secants 166 166 166 166 166 166 166 166 166 166 166 166 166 167 167 167 167 167 Eu. 1 Eu. 1 Eu. 1 Eu. 1 EuA EuA EuA EuA EuA EuA EuA EuA EuA Eu.2 Eu.2 Eu.2 Eu.2 Eu.2 Eu.2 Eu. 1 EuA EuA EuA EuA EuA EuA EuA EuA EuA EuA EuA EuA Eu.2 Eu.2 Eu.2 Eu.2 Eu.2 Eu.2 Eu. I EuA EuA EuA EuA EuA EuA EuA EuA EuA Eu.l EuA EuA EuA EuA EuA EuA EuA Eu I 33 51 17 149 147 197 172 135 Ho. 29 158 127 150 122 150 122 122 122 207 174 17 96 96 5 3 14 172 174 Ho. 29 76 107 99 98 98 98 140 5 Ad. 69 96 96 3 3 4 35 Ca. 3 174 ffo. 41 ffo. 41 Ho. 42 Ad. 69 Ad. 69 37 Ad. 41 Ad. 41 194 A HISTORY OF MATHEMATICAL NOTATIONS 185. OUGHTRED'S MATHEMATICAL SYMBOLS Cont. SYMBOLS MEANINGS or SYMBOLS Clam Mathematical g-w 2 ^8 w'S2 o 1 &-. j|co .4 1 1631 1647 1648 1652 1667 1693 1694 tang C Cent Ho. ' " 9 V T 7 M m m Gr. rain. JNI ! 1 Lo 1 D Tri, tri M X Z cru Zcrur Xcrw Xcrur A L '1 i CO T X Z Tangent .01 of a degree .01 of a degree Degr., min., sec. Hours, min., sec. 180 - angle Equal in no. of degr. ir=3.1418 Canceled* Mean proportion Minus 34 3 4 9*8 -X-=2, --*--=- 23 238 Degree Minute Differentia Aequalia lempore Logarithm Separatrix Differentia Triangle Cent, minute of arc Multiplication" (Z sum, X diff . of sides of rectangle" or triangle Unknown Altit. frust. of pyramid or cone Altit. of part cut off First term ^ Last term .2 No. of terms 1 & Common differ. I g. Sum of all termsj Ho. 29 Ho. 41 Ho. 41 Ho. 42 12 235 236 235 21 20 21 20 21 32 66 07 36 2 6 68 72 100 69 94 66 90 Ar. 1 66 90 Ar. 1 66 90 Ar. 1 99 131 20 32 20 30 19 29 Ho. 23 29 Ho. 23 29 19 45 29 235 Ad. 19 134 68 Ca.2 244 19 237 24 Ca.2 76 191 Eu. 26 70 69 5 17 101 16 16 17 16 38 77 77 13 53 109 109 85,18 85,18 85 85 85,18 51 101 101 80,17 80,17 80 80 80,17 50 99 99 78,16 78,16 78 78 78,16 50 99 99 78,16 78,16 78 78 78,16 50 99 99 78,16 78,16 78 78 78,16 72 141 142 116,26 116,26 116 116 116,26 113 84 19 30,116 30, 116 11 no 30,116 19 INDIVIDUAL WRITERS 195 186. Historical notes 1 to the tables in 181-85: 1. All the symbols, except "Log," which we saw in the 1660 edition of the Circles of Proportion, are given in the editions of 1632 and 1633. 2. In the first half of the seventeenth century the notation for decimal frac- tions engaged the attention of mathematicians in England as it did elsewhere (see 276-89). In 1608 an English translation of Stevin's well-known tract was brought out, with some additions, in London by Robert Norton, under the title, Disme: The Art of Tenths, or, Decimall Arilhmetike ( 276). Steviri's notation is followed also by Henry Lyte in his Art of Tens or Decimall Arith?nelique (London, 1619), and in Johnsons Arithmetick (2d ed.; London, 1633), where 3576.725 is 123 written 3576|725. William Purser in his Compound Interest and Annuities (London, 1634), p. 8, uses the colon (:) as the separator, as did Adrian us Metius in his Geometnae practicae pars I et II (Lvgd., 1625), p. 149, and Rich. Balam in his Algebra (London, 1653), p. 4. The decimal point or comma appears in John Napier's Rabdologia (Edinburgh, 1617). Oughtred's notation for decimals must have delayed the general adoption of the decimal point or comma. 3. This mixture of the old and the new decimal notation occurs in the Key of 1694 (Notes) and in Gilbert Clark's Oughtredus explicatus 2 only once; no reference is made to it in the table of errata of either book. On Oughtred's Opuscula mathc- matica hactenus inedita, the mixed notation 128,57 occurs on p. 193 fourteen times. Oughtred's regular notation 128 [57 hardly ever occurs in this book. We have seen similar mixed notations in the Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster, Sometime publike Professor of Astronomic in Gresham Coltedgc, in London, by John Twysden (London, 1659), p. 13 of the "Observations eclipsi- um"; we find there 32.466, 31.008. 4. The dot (.), used to indicate ratio, is not, as claimed by some writers, used by Oughtred for division. Oughtred does not state in his book that the dot (.) signifies division. We quote from an early arid a late edition of the Clavis. He says in the Clavis of 1694, p. 45, and in the one of 1648, p. 30, "to continue ratios is to multiply them as if they were fractions." Fractions, as well as divisions, are indicated by a horizontal line. Nor does the statement from the Clavis of 1694, p. 20, and the edition of 1648, p. 12, "In Division, as the Divisor is to Unity, so is the Dividend to the Quotient," prove that he looked upon ratio as an indicated division. It does not do so any more than the sentence from the Clavis of 1694, and the one of 1648, p. 7, "In Multiplication, as 1 is to either of the factors, so is the other to the Product," proves that ho considered ratio an indicated multiplica- tion. Oughtred says (Clavis of 1694, p. 19, and the one of 1631, p. 8): "If Two Numbers stand one above another with a Line drawn between them, 'tis as much 12 5 as to say, that the upper is to be divided by the under; as -j- and -^ " 1 N. 1 refers to the Circles of Proportion. The other notes apply to the super- scripts found in the column, "Meanings of Symbols." 2 This is not a book written by Oughtred, but merely a commentary on the Clavis. Nevertheless, it seemed desirable to refer to its notation, which helps to show the changes then in progress. 196 A HISTORY OF MATHEMATICAL NOTATIONS In further confirmation of our view we quote from Oughtred's letter to W. Robinson: ''Division is wrought by setting the divisor under the dividend with a line between them." 1 5. In Gilbert Clark's Oughtredus explicatus there is no mark whatever to sepa- rate the characteristic and mantissa. This is a step backward. 6. Oughtred's language (Clavisoi 1652, p. 21) is: "Ut 7.4: 12.9 vel 7.7-3: 12.12 3. Arithmetic^ proportionates sunk" As later in his work he does not use arith- metical proportion in symbolic analysis, it is not easy to decide whether the sym- bols just quoted were intended by Oughtred as part of his algebraic symbolism or merely as punctuation marks in ordinary writing. Oughtred's notation is adopted in the article "Caractere" of the Encyclopedic methodique (mathematiques) , Paris: Liege, 1784 (see 249). 7. In the publications referred to in the table, of the years 1648 and 1694, the use of : to signify ratio has been found to occur only once in each copy; hence we are inclined to look upon this notation in these copies as printer's errors. We are able to show that the colon (:) was used to designate geometric ratio some years before 1657, by at least two authors, Vincent Wing the astronomer, and a school- master who hides himself behind the initials "R.B." Wing wrote several works. 8. Oughtred's notation A.B::C.D, is the earliest serviceable symbolism for proportion. Before that proportions were either stated in words as was customary in rhetorical modes of exposition, or else was expressed by writing the terms of the proportion in a line with dashes or dots to separate them. This practice was in- adequate for the needs of the new symbolic algebra. Hence Oughtred's notation met with ready acceptance (see 248-59). 9. We have seen this notation only once in this book, namely, in the expres- sion R.S. =3.2. 10. Oughtred says (Clavis of 1694, p. 47), in connection with the radical sign, "If the Power be included between two Points at both ends, it signifies the uni- versal Root of all that Quantity so included; which is sometimes also signified by b and r, as the i/b is the Binomial Root, the -\/r the Residual Root." This notation is in no edition strictly adhered to; the second : is often omitted when all the terms to the end of the polynomial are affected by the radical sign or by the sign for a power. In later editions still greater tendency to a departure from the original notation is evident. Sometimes one dot takes the place of the two dots at the end; sometimes the two end dots are given, but the first two are omitted; in a few instances one dot at both ends is used, or one dot at the beginning and no symbol at the end; however, these cases are very rare and are perhaps only printer's errors We copy the following illustrations: Q : A -E: est Aq-2AE+Eq, for (A -E? = A 2 -2A#-f-# 2 (from Chans of 1631, p. 45) , for (from Clavis of 1648, p. 106) : BA+ CA =BC+ Z), for ^(BA+CA) = BC+D (from Clavis of 1631, p. 40) AB . , ABq CXS . ( AE . I /ZE 2 CXS\ A ,. ni . f i -- R ~ : =4., for -13-+ <v/ (-4 --- --) "A. (from Clavis of 1652, p. 95) 1 Rigaud, Correspondence of Scientific Men of the Seventeenth Century, Vol. I (1841), Letter VI, p. 8. INDIVIDUAL WRITERS 197 Q.Hc+Ch : for (Hc+Ch)* (from Claris of 1652, p. 57) Q.A-X=, for (A-X)*= (from Clavis of 1694, p. 97) --fr.tt. ~!-CD.=A, for|+J(^-CzA =A (from Oughtredus explicates [1682], p. 101) 11. These notations to signify aggregation occur very seldom in the texts re- ferred to and may be simply printer's errors. 12. Mathematical parentheses occur also on p. 75, 80, and 117 of G. Clark's Oughtredus explicates. 13. In the Clavis of 1631, p. 2, it says, "Signum additionis siue affirmationis, est+plus" and "Signum subductionis, siue negationis est minus." In the edition of 1694 it says simply, "The Sign of Addition is + more" and "The Sign of Sub- traction is less," thereby ignoring, in the definition, the double function played by these symbols. 14. In the errata following the Preface of the 1694 edition it says, for "more or mo. r. [ead] plus or pi."', for less or le. r.[ead] minus or mi." 15. Oughtred's Clavis mathematicae of 1631 is not the first appearance of X as a symbol for multiplication. In Edward Wright's translation of John Napier's Descriptio, entitled A Description of the Admirable Table of Logarithms (London, 1618), the letter "X" is given as the sign of multiplication in the part of the book called "An Appendix to the Logarithms, shewing the practise of the calculation of Triangles, etc." The use of the letters x and X for multiplication is not uncommon during the seventeenth and beginning of the eighteenth centuries. We note the following instances: Vincent Wing, Doctrina theorica (London, 1656), p. 63; John Wallis, Arithmetica infinitorum (Oxford, 1655), p. 115, 172; Moore's Arithmelick in two Books, by Jonas Moore (London, 1660), p. 108; Antoine Arnauld, Novveavx elemens de geometrie (Paris, 1667), p. 6; Lord Brounker, Philosophical Transactions, Vol. II (London, 1668), p. 466; Exercitatio geometrica, auctore Laurenlio Lorenzinio, Vincentii Viviani discipulo (Florence, 1721). John Wallis used the X in his Elenchus geometriae Hobbianae (Oxoniae, 1655), p. 23. 16. in as a symbol of multiplication carries with it also a collective meaning; for example, the Clavis of 1652 has on p. 77, "Erit \Z + \B in \Z - \B = \Zq - $Bq." 17. That is, the line AB squared. 18. These capital lettprs precede the expression to be raised to a power. Sel- dom are they used to indicate powers of monomials. From the Clavis of 1652, p. 65, we quote: "C : A +E : +Eq=2Q : \A +E : +2Q4A ," i.e., 19. L and I stand for the same thing, "side" or "root," I being used generally when the coefficients of the unknown quantity are given in Hindu-Arabic numerals, so that all the letters in the equation, viz., I, q, c, qq, qc, etc., are small letters. The Clavis of 1694, p. 158, uses L in a place where the Latin editions use I. 20. The symbol i/u does not occur in the Clavis of 1631 and is not defined in the later editions. The following throws light upon its significance. In the 1631 edition, chap, xvi, sec. 8, p. 40, the author takes i/qBA+ B = CA, gets from it y qBA =CA-B, then squares both sides and solves for the unknown A. He passes 198 A HISTORY OF MATHEMATICAL NOTATIONS next to a radical involving two terms, and says: "Item \/q vniuers : BA+ CA : D BC : vel per transpositionem 1/9 : BA-+-CA =BC+D"' r he squares both sides and solves for A. In the later editions he writes "|/V in place of "i/g vniuers : " 21. The sum Z = A+E and the difference X=A E are used later in imita- tion of Oughtred by Samuel Foster in his Miscellanies (London, 1659), "Of Projection/' p. 8, and by Sir Jonas Moore in his Arithmelick (3d ed.; London, 1688), p. 404; John Wallis in his Operum mathematicorum pars prima (Oxford, 1657), p. 169, and other parts of his mathematical writings. 22. Harriot's symbols > for "greater" and < for "less" were far superior to the corresponding symbols used by Oughtred. 23. This notation for "less than" in the Ho. occurs only in the explanation of "Fig. Efi." In the text (chap, ix) the regular notation explained in En. is used. 24. The symbol GO so closely resembles the symbol <v> which was used by John Wallis in his Operum mathematicorum pars prima (Oxford, 1657), p. 208, 247, 334, 335, that the two symbols were probably intended to be one and the same. It is difficult to assign a good reason why Wallis, who greatly admired Oughtred and was editor of the later Latin editions of his Clavis mathematicae, should purposely reject Oughtred's GO and intentionally introduce ~ as a substi- tute symbol. 25. Von Braunmiihl, in his Geschichte dcr Trigonometric (2. Teil; Leipzig, 1903), p. 42, 91, refers to Oughtred's Trigonomelria of 1657 as containing the earliest use of abbreviations of trigonometric functions and points out that a half- century later the army of writers on trigonometry had hardly yet reached the standard set by Oughtred. This statement must be modified in several respects (see 500-526). 26. This reference is to the English edition, the Trigonometric of 1657. In the Latin edition there is printed on p. 5, by mistake, 8 instead of s versus. The table of errata makes reference to this misprint. 27. The horizontal line was printed beneath the expression that was being crossed out. Thus, on p. 68 of the Clavis of 1631 there is: BGqq-BGqX2BK XBD+BKq XBDg = BGqXBDq+BGq X BKg -BGqX2BKXBD + BGq X 4CAq. 28. This notation, says Oughtred, was used by ancient writers on music, who "are wont to connect the terms of ratios, either to be continued" as in |X| = 2, "or diminished" as in \ -s- j = f . 29. See n. 15. 30. Cru and crur are abbreviations for crurum, side of a rectangle or right tri- angle. Hence Z cru means the sum of the sides, X cru t the difference of the sides. 187. Oughtred's recognition of the importance of notation is voiced in the following passage: ". . . . Which Treatise being not written in the usuall synthetical manner, nor with verbous expressions, but in the inventive way of Analitice, and with symboles or notes of things instead of words, seemed unto many very hard; though indeed it was but their owne diffidence, being scared by the newness of the delivery; and not any INDIVIDUAL WRITERS 199 difficulty in the thing it selfe. For this specious and symbolicall man- ner, neither racketh the memory with multiplicity of words, nor chargeth the phantasie with comparing and laying things together; but plainly presenteth to the eye the whole course and processe of every operation and argumentation/' 1 Again in his Circles of Proportion (1632) , p. 20 : "This manner of setting downe theoremes, whether they be Pro- portions, or Equations, by Symboles or notes of words, is most excel- lent, artificiall, and doctrinall. Wherefore I earnestly exhort every one, that desireth though but to looke into these noble Sciences Mathematicall, to accustome themselves unto it: and indeede it is easie, being most agreeable to reason, yea even to sence. And out of this working may many singular consectaries be drawne: which without this would, it may be, for ever lye hid." ENGLISH: THOMAS HARRIOT (1631) ' 188. Thomas Harriot's Artis analyticae praxis (London, 1631) appeared as a posthumous publication. He used small letters in place of Vieta's capitals, indicated powers by the repetition of factors, and invented > and < for "greater" and "less." Harriot used a very long sign of equality =. The following quo- tation shows his introduction of the now customary signs for "greater" and "smaller" (p. 10): "Comparationis signa in sequentibus vsurpanda. Aequalitatis ut a = b. significet a acqualem ipi b. Maioritatis :r> ut a ;>- 6. significet a maiorem quam b. Minoritatis <d.ut a -<: b significet a minorern quam ft." Noteworthy is the notation for multiplication, consisting of a vertical line on the right of two expressions to be multiplied together of which one is written below the other; also the notation for complex fractions in which the principal fractional line is drawn double. Thus (p. 10): ac ado b b b aaa b aaa ac , d bd * 1 William Oughtred, The Key of the Mathematics (London, 1647), Preface. 200 A HISTORY OF MATHEMATICAL NOTATIONS Harriot places a dot between the numerical coefficient and the other factors of a term. Excepting only a very few cases which seem to be printer's errors, this notation is employed throughout. Thus (p. 60): "Aequationis aaa 3.baa+3.bba=+2.bbb est 2.6, radix radici quaesititiae a. aequalis ." Probably this dot was not intended as a sign of multiplication, but simply a means of separating a numeral from what follows, according to a custom of long standing in manuscripts and early printed books. On the first twenty-six pages of his book, Harriot frequently writes all terms on one side of an equation. Thus (p. 26) : "Posito igitur cdf=aaa. est aaa cdf\ = a+b I Est autem ex genesi aaacdf\ = aaaa-t-baaacdfabcdf. a+b \ quae est aequatio originalis hie designata. Ergo .... aaaa+baaacdfabcdf. = ." Sometimes Harriot writes underneath a given expression the result of carrying out the indicated operations, using a brace, but without using the regular sign of equality. This is seen in Figure 87. The first equation is 52= 3a+aaa, where the vowel a represents the unknown. Then the value of a is given by Tartaglia's formula, as ^26+ V 675+ ^26 -1/675 = 4. Notice thal^VS.)" indicates that the cube root is taken of the binomial 26+1/675. In Figure 88 is exhibited Harriot's use of signs of equality placed vertically and expressing the equality of a polynomial printed above a horizontal line with a polynomial printed below another horizontal line. This exhibition of the various algebraic steps is clever. FRENCH: PIERRE HERIGONE (1634, 1644) 189. A full recognition of the importance of notation and an almost reckless eagerness to introduce an exhaustive set of symbols is exhibited in the Cursus mathematicus of Pierre Hrigone, in six volumes, in Latin and French, published at Paris in 1634 and, in a second edition, in 1644. At the beginning of the first volume is given INDIVIDUAL WRITERS 201 40: ^^*!!3*!^j FIG. 87. From Thomas Harriot's Ar/is analyticae praxis (1631), p. 101 Si daripoffit radix alinua arquatioms radio 4* xquaHs,<iux radicibu$ ^ e. d. ina:- qualis fit , cfto ilia / fiue alia quxcunquc* Pofitoigiwr/=^ mtfffftfff+tttf +ffff Hoceft +ffff'ff/+*ff*fff t, dff Ergo QiJod eft contra Lemmatishypothcfui. Non eft igitur /=='. vtcratpofltum. Quod de alia quacnncjue ex fimilidc- duftione demonftrandom eft. Fia. 88. From Thomas Harriot's Artis analytical praxis (1631), p. 65 202 A HISTORY OF MATHEMATICAL NOTATIONS an explanation of the symbols. As found in the 1644 edition, the list is as follows: + plus is, signifie le plurier ~ minus 2|2 aequalis ~: differentia 3|2 maior < inter se, entrflks 2|3 minor 4 n in, en i tertia pars 4 ntr. inter, entre I quarta pars .11 vel, ou I duae tertiae TT, ad, a a,b, 11 ab rectangulum quod sit 5< pentagonum, penta- ductu A in B gone est punctum 6< hcxagonum est recta linea l/4< latus quadrati <, Z est angulus l/-5< latus pentagon! _J est angulus rectus a2 A quadratum O est circulus a3 A cubus *3> & est pars circumfer- a4 A quadrato-quadratu. entiae circuli et sic infinitum. Q, o est segmentu circuli = parallela A est triangulum JL perpendicularis D est quadratum est nota genitini, sig- a est rectangulum nifie (de) <3> est parallelogrammum ; est nota numeri plural- <0> piped, est parallelepipedum In this list the symbols that are strikingly new are those for equality and inequality, the ^ as a minus sign, the being made to represent a straight line. Novel, also, is the expression of exponents in Hindu- Arabic numerals and the placing of them to the right of the base, but not in an elevated position. At the beginning of Volume VI is given a notation for the aggregation of terms, in which the comma plays a leading role: "O a2~5a+6, a~4: virgula, la virgule, dis- tinguit multiplicatorem a~4 d multiplicado Ergo a 5+4+3, 7~3:~10, est 38." Modern: The rectangle (a 2 5a+6) (a 4) , Rectangle (5+4+3) (7-3) -10 = 38 . "hg TT ga 2|2 hb <* bd, signifi. HO est ad GA, vt HB ad BD ." INDIVIDUAL WRITERS 203 '' '-'-* J ~ * 4i u ^~ i ' J - foufticnt l' , FIG. 89. From P. Herigone, Cursuv mathematicus, Vol. VI (1641) ; proof oi the Pythagorean theorem. 204 A HISTORY OF MATHEMATICAL NOTATIONS Modern: hg : ga = hb : bd . "l/16+9 est 5, se pormoit de serirc plus dis- tinctement ainsi , l/-(16+9) 111AT6+9, est 5:i/-9, +4, sont 7: j/-9, +|/-4 sont 5: " Modern: y 16+9 is 5, can be written more clearly thus, V 7 - (16+9) or v 7 - 16+9, is 5; ]/-9, +4, are 7; 1/-9, + i/-4 are 5 . FRENCH: JAMES HUME (1635, 1636) 190. The final development of the modern notation for positive integral exponents took place in mathematical works written in French. Hume was British by birth. His Le traite d'algebre (Paris, 1635) contains exponents and radical indexes expressed in Roman numerals. In Figure 90 we see that in 1635 the plus (+) and minus ( ) signs were firmly established in France. The idea of writing exponents without the bases, which had been long prevalent in the writings of Chuquet, Bombelli, Stevin, and others, still prevails in the 1635 publication of Hume. Expressing exponents in Roman symbols made it possible to write the exponent on the same line with the coeffi- cient without confusion of one with the other. The third of the ex- amples in Figure 90 exhibits the multiplication of 8x 2 +3x by IQx, yielding the product 80x 3 +30x 2 . The translation of part of Figure 91 is as follows: "Example: Let there be two numbers 1/9 and 1/8, to reduce them it will be necessary to take the square of 1/8, because of the II which is with 9, and the square of the square of 1/9 and you obtain 1/6561 and 1/64 f8 to 1/64 1/9 to 1/729 1/3 to 1/8 [should be 1/9] 1/2 to 1/9 [should be 1/8] 1/3 to 1/9 1/2 to V32 ." The following year, Hume took an important step in his edition of Ualgebre de Viete (Paris, 1636), in which he wrote A iu for A 3 . Except for the use of the Roman numerals one has here the notation used by Descartes in 1637 in his La geometric (see 191). INDIVIDUAL WRITERS 205 FRENCH : RENE DESCARTES (1637) 191. Figure 92 shows a page from the first edition of Descartes' La gtomttrie. Among the symbolic features of this book are: (1) the use of small letters, as had been emphazised by Thomas Harriot; I ^ ' Jrr ^ f* ' * ' * l 1* f> ^S-J ' ' FIG. 90. Roman numerals for unknown numbers in James Huine, Algbbre (Paris, 1635). (2) the writing of the positive integral exponents in Hindu-Arabic numerals and in the position relative to the base as is practiced today, 206 A HISTORY OF MATHEMATICAL NOTATIONS except that aa is sometimes written for a 2 ; (3) the use of a new sign of equality, probably intended to represent the first two letters in the word aequalis, but apparently was the astronomical sign, & taurus, tf^V f" V , >**. Eir Lv?,,' 1 ltrt$efoh 1'jG. 91. -Radicals in Junics Hume, Algvbrc (1635) INDIVIDUAL WRITERS 207 . jtfteFW^la^ T'W ^-J ^s%'^^>l& $y^ f fa#WS'<;\'&fck WV" ">' '^-- ';' "MV& ^ ; V^;'ji4^ ; ?^iHe;|t;^;Wtt^^ ri.-ft -i'W^ FIG. 92. A page from Ren6 Descartes, Lo gtamttrie (1637) 208 A HISTORY OF MATHEMATICAL NOTATIONS placed horizontally, with the opening facing to the left; (4) the uniting of the vinculum with the German radical sign j/, so as to give i/*"~~, an adjustment generally used today. The following is a quotation from Descartes' text (ed., Paris, 1886, p. 2): "Mais souvent on n'a pas besoin de tracer ainsi ces lignes sur le papier, et il suffit de les designer par quelques lettres, chacune par une seule. Comme pour ajouter le ligne BD a GH, je nomme Pune a et Pautre 6, et cris a+b; et a b pour soustraire b de a; et ab pour les multiplier Pune par Pautre; et T pour diviser a par b; et aa ou a 2 pour multiplier a par soi-meme; et a 3 pour le multiplier encore une fois par a, et ainsi a Pinfini. " The translation is as follows: "But often there is no need thus to trace the lines on paper, and it suffices to designate them by certain letters, each by a single one. Thus, in adding the line BD to GH, I designate one a and the other 6, and write a+b; and a 6 in sub- tracting b from a; and ab in multiplying the one by the other; and j- in dividing a by 6; and aa or a 2 in multiplying a by itself; and a 3 in multiplying it once more again by a, and thus to infinity/' ENGLISH: ISAAC BARROW (1655, 1660) 192. An enthusiastic admirer of Oughtred's symbolic methods was Isaac Barrow, 1 who adopted Oughtred's symbols, with hardly any changes, in his Latin (1655) and his English (1660) editions of Euclid, Figures 93 and 94 show pages of Barrow's Euclid. ENGLISH: RICHARD RAWLINSON (1655-68) 193. Sometime in the interval 1655-68 Richard Rawlinson, of Oxford, prepared a pamphlet which contains a collection of litho- graphed symbols that are shown in Figure 95, prepared from a crude freehand reproduction of the original symbols. The chief interest lies in the designation of an angle of a triangle and its opposite side by the same letter one a capital letter, the other letter small. This simple device was introduced by L. Euler, but was suggested many years earlier by Rawlinson, as here shown. Rawlinson designated spherical 1 For additional information on his symbols, see 456, 528. INDIVIDUAL WRITERS 209 triangles by conspicuously rounded letters and plane triangles by letters straight in part. i' ' , - -< i - ' ~ &raW:'fit tiM 1 ' '?^ p^p r ^. w j _ T _ *f"(L T W r^ ib'ii.tiffOTMafflfK ijj^ik^ ^ i 1 j - _ .,.,.----"-- I- *y8p ' At>l4<i)w4*w ABo^>fififl' rfe/wwr /gr4 ^^.fe^W 11 */^? * w^B-AD, DB/^ifrf/*f- **r"^^*uJ^" (3 ' fl ^ w ^^ M> <S*^ SM / ! ' AB taxitftHt '?*~H~~vtW$to ?***>* <5K ,' Kr4(W fj? qtoetMuto '&'-. * ?JBgtM^fe f D' JM tevtu&ae tammta^m: 5*rsT#.ji !*w-l^fcfe .^/ff* : e^.it?* gj; l**fiW : ^^^^V^(i^r4l^^' fW' <w< /* *& X,;^' W>" fWtlf i/^f.^ ,*3^ *^f - .. i . ; M , ,,j!^ li u.^- A- *i, ^ llW i)flr<a( AB q$Mt*t t ^'k$$y*,^&: Atypfrim* ^14'A^l^ 1 , 1^^I^' % ' ' r i i K*i t*i k, I IT- * *_.,tr FIG. 93. Latin edition (1655) of Barrow's Euclid. Notes by Isaac Newton. (Taken from Isaac Newton: A Memorial Volume [ed. W. J. Greenstreet; London, 1927], p. 168.) 210 A HISTORY OF MATHEMATICAL NOTATIONS DF :: CB FH,vth 3 t , ,:J*:^':'; ;u ,f\ '| y 1 ' ; /y^y/v^L; ^y^^V^'^r' , V J AS^^AIJi^ iJ^Jl^'"^^^ ^ |f[ ! -' ' '' ^ ^W^CBLF^fU^f^'^^H:;, -- ---tin ^;ptt^3fc*|;%' ^:; 4 FIG. 94. English edition of Isaac Barrow's Euclid INDIVIDUAL WRITERS 211 SWISS! JOHANN HEINRICH RAHN (1659) 194. Rahn published in 1659 at Zurich his Teutsche Algebra, which was translated by Thomas Brancker and published in 1668 at London, with additions by John Pell. There were some changes in the symbols as indicated in the following comparison : Meaning German Edition, 1659 English Edition, 1668 1 Multiplication * (p 7) Same (r> ft) 2 d~\-b times a b 2 + hl (P H) Same (p 12) 3. Division 4. Cross-multiplication 5 Involution -J- (P. 8) *X (p. 25) Archimedean spi- Same *X Ligature of (p. 7) (p. 23) omicron and 6 Evolution ral (Fig. 96) (p. 10) Ligature of two sigraa (Fig. Same 97) (p. 9) fn 0^ 7. Erf Ull ein quadrat \ Compleat the square / epsilons(Fig.96)(p. 11) #D (P 16) CD (P- 14) 8 Sixth root / /aaa V cubo -cubic k V o/aaa V (p. 32) 9 Therefore y/' \ aa = Vc.a (p. 34) .'. (usually) (p 53) cubo-cubick V '.' (usually) of aa= jc.a (n 37^ 10. Impossible (absurd) 11. Equation expressed in an- other way 2 (P- 01) (p. 67) 01 Same (P. 48) (p 64) 12. Indeterminate, "liberty of as- suming an equation" 13. Nos. in outer column refer- ring to steps numbered in middle column 14. Nos in outer column not re- ferring to numbers in middle column (*) (p. 89^ 1,' 2,' 3*. etc. (p. 3) 1 2, 3, etc (p 3) Same 1, 2, 3, etc. 1, 2, 3, etc (P. 77) (p. 3) (p 3) REMARKS ON THESE SYMBOLS No. 1. Rahri's sign * for multiplication was used the same year as Brancker's translation, by N. Mercator, in his Louarithrnotechnia (London, 1668), p. 28. No. 4. If the lowest common multiple of abc and ad is required, Rahn writes -T .; then -,'-*X-7 yields abed in each of the two cross-multiplications. ad d No. 8. Hahn's and Brancker's modes of indicating the higher powers and roots differ in principle and represent two different procedures which had been competing for supremacy for several centuries. Rahn's V^- means the sixth root, 2X3 = 6, and represents the Hindu idea. Brancker's cubo-cubick root means the "sixth root," 3+3 = 6, and represents the Diophantine idea. No. 9. In both editions occur both /. and v, but /. prevails in the earlier edition; v prevails in the later. No. 10. The symbols indicate that the operation is impossible or, in case of a root, that it is imaginary. Wo. 11. The use of the comma is illustrated thus: The marginal column (1668, p. 54) gives "6, 1," which means that the sixth equation "Z = A" and the first equation "A=6" yield Z=6. No. 12. For example, if in a right triangle h, b, c, we know only b c, then one of the three sides, say c, is indeterminate. Page 73 of Rahn's Teutsche Algebra (shown in Fig. 96) shows: (1) the first use of -5- in print, as a sign of division; (2) the Archimede- an spiral for involution; (3) the double epsilon for evolution; (4) the 212 A HISTORY OF MATHEMATICAL NOTATIONS use of capital letters J5, D, E, for given numbers, and small letters a, 6, for unknown numbers; (5) the ^ for multiplication; (6) the first use of .'. for "therefore"; (7) the three-column arrangement of which the left column contains the directions, the middle the numbers of ^ * y b , r i ^ / " / 1'iG. 95. Freehand reproduction of Richard Rawlinson's symbols the lines, the right the results of the operations. Thus, in line 3, we have "line 1, raised to the second power, gives aa+2ab-\-bb=DD." ENGLISH: JOHN WALLIS (1655, 1657, 1685) 195. Wallis used extensively symbols of Oughtred and Harriot, but of course he adopted the exponential notation of Descartes (1637). Wallis was a close student of the history of algebra, as is illustrated INDIVIDUAL WRITERS 213 by the exhibition of various notations of powers which Wallis gave in 1657. In Figure 98, on the left, are the names of powers. In the first column of symbols Wallis gives the German symbols as found in Stifel, which Wallis says sprang from the letters r, z, c, J, the first b' t*-- 1 iflY'^V^'' * r 'fT^fl' u?'^; ^fy^W^''' 1 ^ $X'''$**'1'-.i ?* '^'"^ ^ r'l; '^ - "'p^-'r 'LjX'V^ ^ ;'j'^ ; /^v'"^, 'i n 4' 1 ;'- ;!;.'--' ^ ^ --<l : , ^ '7^^^''' : ''Mi^_i^^:l^.: FIG. 90. From Rahn, Teutsche Algebra (1659) letters of the words res, zensus, cubus, sursolidus. In the second column are the letters R, Q, C, S and their combinations, Wallis remarking that for R some write N; these were used by Vieta in numerical equa- tions. In the third column are Vieta's symbols in literal algebra, as abbreviated by Oughtred; in the fourth column Harriot's procedure is indicated; in the fifth column is Descartes' exponential notation. 214 A HISTORY OF MATHEMATICAL NOTATIONS In his Arithmetica infinitorum 1 he used the colon as a symbol for for 1/aD-a 2 ; aggregation, as i/:a?+l for T/a 2 +l, \/:oDa*: Oughtred's notation for ratio and proportion, -fr for continued pro- portion. As the sign for multiplication one finds in this book X and X, both signs occurring sometimes on one and the same page (for instance, p. 172). In a table (p. 169) he puts D for a given number: "Verbi gratia; si numerus hac nota D designatus supponatur cognitus, reliqui omnes etiam cognoscentur." It is in this book and in his De 6l *=? } . j 2 4*7 3 5 FIG. 97. From Braiicker's translation of Halm (1668). The same arrange- ment of the solution as in 1659, but the omicron-sigma takes the place of the Archimedean spiral; the ordinal numbers in the outer column are not dotted, while the number in that column which does not refer to steps in the middle column carries a bar, 2. Step 5 means ''line 4, multiplied by 2, gives 4ab = 2DD 27V' sectionibus conicis that Wallis first introduces oo for infinity. He says (p. 70) : "Cum enim primus terminus in serie Primanorum sit 0, primus terminus in serie reciproca erit oo vel infinitus : (sicut, in divisione, si diviso sit 0, quotiens erit infinitus)"; on pages 152, 153: " . . . . quippe ^- (pars infinite parva) habenda erit pro nihilo," "oo X-&-B = B" "Nam oo, oo +1 oo 1, perinde sunt"; on page 168: "Quamvis enim oo XO non aliquem determinate numerum designet. . . . ." An imitation of Oughtred is Wallis 7 "HT:1|-|," which occurs in 4 his famous determination by interpolation of as the ratio of two in- 4 finite products. At this place he represents our by the symbol D. 7T 1 Johannis Wallisii Arithmetica infinitorum (Oxford, 1655). INDIVIDUAL WRITERS 215 He says also (p. 175) : "Si igitur ut j/ : 3 X 6 : significat terminum medi- um inter 3 et 6 in progressione Geometrica aequabili 3, 6, 12, etc. (continue multiplicando 3X2X2 etc.) ita ]7T : 1 1| : significet terminum medium inter 1 et f in progressione Geometrica decrescente 1, f, *-, etc. (continue multiplicando iXfXf, etc.) erit D=)?r:l|f: Et propterea circulus est ad quadratum diametri, ut 1 ad r:l|f." He uses this symbol again in his Treatise of Algebra (1685), pages 296, 362. 7* Dt 'Witifiwif Al&dticA* ' CAP. ri^ -' Pole ft <ts (tu Radix 3? TL ^ A * a r Quadratutn 7^ S^ s Aq ** a* 2 Cubus < CT" Ac aaa a * ' 3 Q^ud. quadratum 2^ ^^ Aqq -^^4 <J A ^ Sufdcfolidtim f^ S" Aqc 8cc. tf l 5 QiiadiCubi; y 5P Acc tf * * ' 3L m Surdtfolidam. B|o bS Aqqc ^^ y Quad, quad.quad, ^ ^%%. &&!- ^qcc 4 1 '8 Cubicubut << CC Accc J* 9 Quad, Surdcfol. 2^/<? ^S Aqqcc J '-* 10 3 m Surdcfol idum C/tf cS Aqccc 4"- H Qiiad. quad. cnbi ( 2^7j^ <^^p Acccc .*'* 12 4 m Surdrfolidum D/' d S Aqqccc rf i$ 13 Qtiad. 2* Surdcfol. - %B[# ^J 3 ^ Aqcccc <i >A i.j, Cubui Surdcfol. *pf^. CS Accccc a ^ i<? Quad. quad quad. quad. FIG. 98. From John Wallis, Operum mathematicorum pars prima (Oxford, 1657), p. 72. The absence of a special sign for division shows itself in such pas- sages as (p. 135): "Ratio rationis hujus ^-~ ad illam , puta /LJ - )- ( ,erit " He uses Oughtred's clumsy notation for decimal ^/ 2CU \LH fractions, even though Napier had used the point or comma in 1617. On page 166 Wallis comes close to the modern radical notation; he writes ' *\/*R" for l/R. Yet on that very page he uses the old designa- tion "i/qqR" for 216 A HISTORY OF MATHEMATICAL NOTATIONS His notation for continued fractions is shown in the following quotation (p. 191): "Esto igitur fractio ejusmodi - 6 continue fracta quaelibet, sic a @ - 5? e designata, 5 ~ > e ^ c< > where The suggestion of the use of negative exponents, introduced later by Isaac Newton, is given in the following passage (p. 74) : "Ubi autem series directae indices habent 1, 2, 3, etc. ut quae supra seriem Aequalium tot gradibus ascendunt; habebunt hae quidem (illis re- ciprocae) suos indices contrarios negativos 1, 2, 3, etc. tanquam tot gradibus infra seriem Aequalium descendentes." In Wallis' Mathesis universalis, 1 the idea of positive and negative integral exponents is brought out in the explanation of the Hindu- Arabic notation. The same principle prevails in the sexagesimal nota- tion, "hoc est, minuta prima, secunda, tertia, etc. ad dextram de- scendendo," while ascending on the left are units "quae vocantur Sexagena prima, secunda, tertia, etc. hoc modo. \\\\ \\\ \\ \ o / // /// //// 49, 36, 25, 15, 1, 15, 25, 36, 49. " That the consideration of sexagesimal integers of denominations of higher orders was still in vogue is somewhat surprising. On page 157 he explains both the "scratch method" of dividing one number by another and the method of long division now current, except that, in the latter method, he writes the divisor underneath the dividend. On page 240: "A, M, V jf " for arithmetic proportion, i.e., to indicate M A = VM. On page 292, he introduces a general root d in this manner: "\/ d R d = R." Page 335 contains the following interesting combination of symbols: / s * " , * * In Modern Symbols "Si A B C : a 7 If A:B = a:p, ' d_ :: ^Tr-" and #:C = /3:7, Erit A C :: a 7." thenA:C = ai7. 196. In the Treatise of Algebra? (p. 46), Wallis uses the decimal point., placed at the lower terminus of the letters, thus: 3.14159, 1 Johannis Wallisii Mathe&ia universalis: sive, Arithmeticum opus integrum (Oxford, 1657), p. 65-68. 2 Op. oil. (London, 1685). INDIVIDUAL WRITERS 217 26535 ..... , but on page 232 he uses the comma, "12,756," ",3936." On page 67, describing Oughtred's Clavis maihematicae y Wallis says: "He doth also (to very great advantage) make use of several Ligatures, or Compendious Notes, to signify the Summs, Differences, and Rec- tangles of several Quantities. As for instance, Of two quantities A (the Greater, and E (the Lesser,) the Sum he calls Z, the Difference X, the Rectangle M ..... " On page 109 Wallis summarizes various practices: "The Root of such Binomial or Residual is called a Root universal; and thus marked \/u y (Root universal,) or j/6, (Root of a Binomial,) or j/r, (Root of a Residual,) or drawing a Line over the whole Compound quantity; or including it (as Oughtred used to do) within two colons; or by some other distinction, whereby it may ap- pear, that the note of Radicality respects, not only the single quantity next adjoining, but the whole Aggregate. As j/6 : 2+1/3-j/r : 2 On page 227 Wallis uses Rahn's sign -f- for division; along with the colon as the sign of aggregation it gives rise to oddities in notation like the following: "ll-2laa+a*: + bb." On page 260, in a geometric problem, he writes "QAE" for the square of the line AE; he uses fp for the absolute value of the difference. On page 317 his notation for infinite products and infinite series is as follows: etc." etc." " for V2- on page 322: On page 332 he uses fractional exponents (Newton having intro- duced the modern notation for negative and fractional exponents in 1676) as follows: V 5 :c 5 +c 4 :r-z 5 : or The difficulties experienced by the typesetter in printing fractional exponents are exhibited on page 346, where we find, for example, "d\ x%" for d*x*. On page 123, the factoring of 5940 is shown as follows: "11)5)3)3)3)2)2) 5940 (2970(1485(495(165(55(11(1 ." 218 A HISTORY OF MATHEMATICAL NOTATIONS In a letter to John Collins, Wallis expresses himself on the sign of multiplication: "In printing my things, I had rather you make use of Mr. Oughtred's note of multiplication, X, than that of $; the other being the more simple. And if it be thought apt to be mistaken for X, it may [be] helped by making the upper and lower angles more obtuse ixj." 1 "I do not understand why the sign of multiplication X should more trouble the convenient placing of the fractions than the other signs + - = > ::." 2 Wallis, in presenting the history of algebra, stressed the work of Harriot and Oughtred. John Collins took some exception to Wallis' attitude, as is shown in the following illuminating letter. Collins says: 3 "You do not like those words of Vieta in his theorems, ex adjunctione piano solidi, plus quadrato quadrati, etc., and think Mr. Oughtred the first that abridged those expressions by symbols; but I dissent, and tell you 'twas done before by Cataldus, Geysius, and Camillus Gloriosus, 4 who in his first decade of exercises, (not the first tract,) printed at Naples in 1627, which was four years before the first edition of the Clavis, proposeth this equation just as I here give it you, viz., lccc+ IQqcc+llqqc- 2304cc- 18364gc - 133000^ - 54505c + 3728q + 8064^ aequatur 4608, finds N or a root of it to be 24, and composeth the whole out of it for proof, just in Mr. Oughtred's symbols and method. Cataldus on Vieta came out fifteen years before, and I can- not quote that, as not having it by me And as for Mr. Ought- red's method of symbols, this I say to it; it may be proper for you as a commentator to follow it, but divers I know, men of inferior rank that have good skill in algebra, that neither use nor approve it Is not A b sooner wrote than Aqcf Let A be 2, the cube of 2 is 8, which squared is 64: one of the questions between Magnet Grisio and Gloriosus is whether 64 = A cc or A qc . The Cartesian method tells you it is A 6 j and decides the doubt." EXTRACT FROM ACTA ERUDITORUM 5 197. "Monendurn denique, nos in posterum in his Actis usuros esse Signis Leibnitianis, ubi cum Algebraicis res nobis fuerit, ne typothetis 1 John Wallis to John Collins, July 21, 1668 (S. P. Rigaud, Correspondence of Scientific Men of the Seventeenth Century, Vol. II [Oxford, 1841], p. 492). 2 Wallis to Collins, September 8, 1668 (ibid., p. 494). 3 Letter to John Wallis, about 1667 (ibid., p. 477-80). 4 " fixer citationum Mathematicarum Decas prima, Nap. 1627, and probably Cataldus' Transformatio Geometrica, Bonon. 1612." 5 Taken from Ada eruditorum (Leipzig, 1708), p. 271. INDIVIDUAL WRITERS 219 tacdia & molestias gratis creemus, utque ambiguitatcs evitemus. Loco igitur lineolae characteribus supraducendae parenthcsin ad- hibebimus, imrno in multiplicatione simplex comma, ex. gr. loco Vaa+bb scribemus V(aa+bb) & pro aa+bbXc ponemus aa+bb, c. Divisionem designabimus per duo puncta, nisi peculiaris quacdam circumstantia morem vulgarem adhiberi suaserit. Ita nobis erit a:& = . Et hinc peculiaribus signis ad denotandam proportionem nobis non erit opus. Si enim f uerit ut a ad 6 ita c ad d, erit a:b c:d. Quod potentias attinet, aa+bb designabimus per (aa+bb) m : unde & Vaa+bb erit (aa+66) 1 : m & Vaa+bb n =(aa+bb) n:m . Nulli vero dubitamus fore, ut Geometrae omnes Acta haec legentes Signorum Leibnitianorum praestantiam animadvertant, & nobiscum in eadem consentiant." The translation is as follows: "We hereby issue the reminder that in the future we shall use in these Acta the Leibnizian signs, where, when algebraic matters concern us, we do not choose the typographi- cally troublesome and unnecessarily repugnant, and that we avoid ambiguity. Hence we shall prefer the parenthesis to the characters consisting of lines drawn above, and in multiplication by all means simply the comma; for example, in place of I/ aa+bb we write l/(aa+W) and for aa+bbXc we take aa+bb t c. Division we mark with two dots, unless indeed some peculiar circumstance directs ad- herence to the usual practice. Accordingly, we have a:6 = r. And it is not necessary to denote proportion by any special sign. For, if a is to b as c is to d, we have a:b = c:d. As regards powers, aa+bb m , we designate them by (aa+bb) m ; whence also V aa+bb becomes m/~ " = (aa+W>) 1:m and ^aa+bb n = (aa+bb) n:m . We do not doubt that all geometers who read the Acta will recognize the excellence of the Leibnizian symbols and will agree with us in this matter." EXTRACT FROM MISCELLANEA BEROLINENSIA 1 198. "Monitum De Characteribus Algebraicis. Quoniam variant Geometrae in characterum usu, nova praesertim Analysi inventa; quae res legentibus non admodum provectis obscuritatem parit; ideo e re visum est exponere, quomodo Characteres adhibeantur Leibnitiano more, quern in his Miscellaneis secuturi sumus. Literae 1 Taken from Miscellanea Berolinensia (1710), p. 155. Article due to G. W, Leibniz. 220 A HISTORY OF MATHEMATICAL NOTATIONS minusculae o, 6, x, y solent significare magnitudines, vel quod idem est, numeros indeterminatos: Majusculae verb, ut A, B, X, Y puncta figurarum; ita ab significat factum ex a in 6, sed AB rectam & puncto A ad punctum B ductam. Huic tamen observationi adeo alligati non sumus, ut non aliquando minusculas pro punctis, majusculas pro numeris vel magnitudinibus usurpemus, quod facile apparebit ex modo adhibendi. Solent etiam literae priores, ut a, 6, pro quantitati- bus cognitis vel saltern determinatis adhiberi, sed posteriorcs, ut x, y, pro incognitis vel saltern pro variantibus. "Interdum pro literis adhibentur Numeri, sed qui idem significant quod literae, utiliter tamen usurpantur relationis exprimendae gratia. Exempli causa: Sint binae aequationes generales secundi gradus pro incognita, x; eas sic exprimere licebit: 10xx> }-*llx> -12 = & 2Qxx> { 2l> 122 = ita in progressu calculi ex ipsa notatione apparet quantitatis cujusque relatio; nempe 21 (ex. gr.) per notam dextram, quae est 1 agnoscitur esse coefficiens ipsius x simplicis, at per notam sinistram 2 agnoscitur esse ex. aeq. secunda: sed et servatur lex quaedam homogeneorum. Et ope harum duarum aequationum tollendo x, prodit aequatio, in qua similiter sc haberc oportet 10, 11, 12 et 12, 11, 10; item 20, 21, 22 et 22, 21, 20; et dcniquc 10, 11, 12 se habent ut. 20, 21, 22. id est si pro 10, 11, 12 substituas 20, 21, 22 et vice versa manet eadem aequatio; idemque est in caeteris. Tales numeri tractantur ut literae, veri autern numeri, discriminis causa, parenthesibus includuntur vel aliter discernuntur. Ita in tali sensu 11.20. significat numeros indefinites 11 et 20 in se invicem ductos, non vero significat 220 quasi esscnt Numeri veri. Sed hie usus ordinarius non est, rariusque adhibetur. "Signa, Additionis nimirum et Subtractions , sunt > I- plus, minus, > J- plus vel minus, > j . priori oppositum minus vel plus. At ( H) vel ( ... | . ) est nota ambiguitatis signorum, independens a priori; et (( {-) vel (( j .) alia independens ab utraque; Differt autern Signum ambiguum a Differentia quantitatum, quae etsi aliquan- do incerta, non tamen ambigua est Sed differentia inter a et b y significat a fe, si a sit majus, et 6 a si b sit majus, quod etiam ap- pellari potest moles ipsius a 6, intelligendo (exempli causa) ipsius 1-2 et ipsius 2 molem esse eandem, nempe -2; ita si a 6 vocemus c utique mol. c, seu moles ipsius c erit 1-2, quae est quan- titas affirmativa sive c sit affirmativa sive negativa, id est, sive sit c idem quod I- 2, sive c sit idem quod 2. Et quantitates duae diversae eandem molem habentes semper habent idem quadratum. INDIVIDUAL WRITERS 221 "Multiplicationem plerumque signifare content! sumus per nudam appositionem: sic ab significat a multiplicari per 6, Numeros multi- plicantes solemus praefigere, sic 3a significat triplum ipsius a interdum tamen punctum vel comma interponimus inter multiplicans et multiplicandum, velut cum 3, 2 significat 3 multiplicari per 2, quod facit 6, si 3 et 2 sunt nurneri veri; et AB, CD significat rectam AB duci in rectam CD, atque inde fieri rectangulum. Sed et commata inter- dum hoc loco adhibemus utiliter, velut a, b*%*c, vel AB, CD k~EF, id est, a duci in 6 f-c, vel AB in CD }-EF; sed de his mox, ubi de vinculis. Porro propria Nota Multiplicationis non solet esse neces- saria, cum plerumque appositio, qualem diximus, sufficiat. Si tamen utilis aliquando sit, adhibebitur potius r\ quam M , quia hoc ambigui- tatem parit, et ita AB/^CD significat AB duci in CD. "Diviso significatur interdum more vulgari per subscriptionem diuisoris sub ipso dividendo, intercedente linea, ita a dividi per 6, significatur vulgo per 7 ; plerumque tamen hoc evitare praestat, efficereque, ut in eadem linea permaneatur, quod sit interpositis duobus punctis; ita ut a: 6 significat a dividi per 6. Quod si a: b rursus dividi debeat per c, poterimus scribere a : b, : c, vel (a : &) : c. Etsi enim res hoc casu (sane simplici) facile aliter exprimi posset, fit enim a : (be) vel a : be non tamen semper divisio actu ipse f acienda cst, sed saepe tantum indicanda, et tune praestat operationis dilatae pro- cessum per commata vel parentheses indicari ..... Et exponens inter- dum lineolis includitur hac modo (T](AjB l-BC) quo significatur cubus rectac AB - ^BC ..... a + n et utiliter interdum lineola sub- ducitur, ne literae exponentiales aliis confundantur; posset etiam scribi fe+n^ a ..... " ____ itav/(a 3 ) vel i/[j[](a 3 ) rursus est a, .... sed f/Z vel i/02 significat radicern cubicam ex eodem numero, et -fr'2 vel i/ Q 2 signifi- cat, radicem indeterminati gradus e ex 2 extrahendam ..... "Pro vinculis vulgo solent adhiberi ductus linearum; sed quia lineis una super alia ductis, saepe nimium spatii occupatur, aliasque ob causas commodius plerumque adhibentur commata et parentheses. Sic a, b+f^c idem est quod a, 6*4 <c vel a(6-J <c); et a-\ <b, idem quod a-f <b, c*-J <d vel (a -4 <) (c-J <), id est, multiplicatum per c~{ d. Et similiter vincula in vin- culis exhibentur. Ita a, 6c-{ <e/H <g etiam sic exprimetur, Et a, be -4 c e/-J <g+i <Wm, n potest etiam 222 A HISTORY OF MATHEMATICAL NOTATIONS sic exprimi: -J <(a(bc> l^e(f+g))+hl?n)n. Quod de vinculis multi- plicationis, idem intelligi potcst de vinculis divisionis, exempli gratia b T e * I . .. , . r - -I < 7-1 , sic scnbetur in una Imea c n nihilque in his difficultatis, modo teneamus, quicquid parenthesin aliquam implet pro una quantitate haberi, .... Idemque igitur locum habet in vinculis cxtractionis radicalis. Sic \a 4 -! <^ e,/*-f <<7 idem est quod i/(a 4 Et pro i/aa -I < b|/cc H < dd e J < V'fi/gg I c AA I < fcfc scribi poterit j/ (aa -f < 6 j/(cc -J < dd)) : , itaque a = 6 significat, a, esse equale ipsi 6, et a=^6 significat a esse majus quam 6, et a =b significat a esse minus quarn b. "Sed et proportionalitas vel analogia de quantitatibus enunciatur, id est, rationis identitas, quam possumus in Calculo exprimere per notam aequalitatis, ut non sit opus pcculiaribus notis. Itaqua a esse ad 6, sic ut I ad m, sic exprimere poterimus a:b = l: m, id est y = . Nota continue proportionalium erit -H-, ita ut -H- a.6.c. etc. sint con- tinue proportionales. Interdum nota Similitudinis prodest, quae est c^ , item nota similitudinis aequalitatis simul, seu nota congruitatis & 9 Sic DEF v> PQR significant Triangula haec duo esse similia; at DEF & PQR significant congruere inter se. Huic si tria inter se habeant eandem rationem quam tria alia inter se, poterimus hoc exprimere nota similitudinis, ut a; 6; a> Z; w; n quod significat esse a ad 6, ut I ad m, et a ad c ut I ad n, et b ad c ut m ad n ..... " The translation is as follows: "Recommendations on algebraic characters. Since geometers differ in the use of characters, especially those of the newly invented anal- ysis, a situation which perplexes those followers who as yet are not very far advanced, it seems proper to explain the manner of using the characters in the Leibnizian procedure, which we have adopted in the INDIVIDUAL WRITERS 223 Miscellanies. The small letters a, &, x, y, signify magnitudes, or what is the same thing, indeterminate numbers. The capitals on the other hand, as A, 5, X, F, stand for points of figures. Thus ab signifies the result of a times 6, but AB signifies the right line drawn from the point A to the point B. We are, however, not bound to this convention, for not infrequently we shall employ small letters for points, capitals for numbers or magnitudes, as will be easily evident from the mode of statement. It is customary, however, to employ the first letters a, 6, for known or fixed quantities, and the last letters x, y, for the un- knowns or variables. "Sometimes numbers are introduced instead of letters, but they signify the same as letters; they are convenient for the expression of relations. For example, let there be two general equations of the second degree having the unknown x. It is allowable to express them thus: Hte+ llz+ 12 = and 2(te+21z+22 = 0. Then, in the prog- ress of the calculation the relation of any quantity appears from the notation itself; thus, for example, in 21 the right digit which is 1 is recognized as the coefficient of x, and the left digit 2 is recognized as belonging to the second equation; but also a certain law of homo- geneity is obeyed. And eliminating x by means of these two equa- tions, an equation is obtained in which one has similarity in 10, 11, 12 and 12, 11, 10; also in 20, 21, 22 and 22, 21, 20; and lastly in 10, 11, 12 and 20, 21, 22. That is, if for 10, 11, 12, you substitute 20, 21, 22 and vice versa, there remains the same equation, and so on. Such numbers are treated as if letters. But for the sake of distinction, they are in- cluded in parentheses or otherwise marked. Accordingly, 11-20. signifies the indefinite numbers 11 and 20 multiplied one into the other; it does not signify 220 as it would if they were really numbers. But this usage is uncommon and is rarely applied. "The signs of addition and subtraction are commonly + plus, minus, plus or minus, T the opposite to the preceding, minus or plus. Moreover ( ) or ( + ) is the mark of ambiguity of signs that are independent at the start; and (() or (( + ) are other signs inde- pendent of both the preceding. Now the symbol of ambiguity differs from the difference of quantities which, although sometimes unde- termined, is not ambiguous But a b signifies the difference between a and b when a is the greater, b a when 6 is the greater, and this absolute value (moles) may however be called itself a 6, by understanding that the absolute value of +2 and 2, for example, is the same, namely, +2. Accordingly, if a b is called c, thenraoZ. c or the absolute value of c is +2, which is an affirmative quantity whether 224 A HISTORY OF MATHEMATICAL NOTATIONS c itself is positive or negative; i.e., either c is the same as +2, or c is the same as 2. Two different quantities having the same absolute value have always the same square. "Multiplication we are commonly content to indicate by simple apposition: thus, ab signifies a multiplied by 6. The multiplier we are accustomed to place in front; thus 3a means the triple of a itself. Sometimes, however, we insert a point or a comma between multi- plier and multiplicand; thus, for example, 3,2 signifies that 3 is multi- plied by 2, which makes 6, when 3 and 2 are really numbers; and AB,CD signifies the right line AB multiplied into the right line CD, producing a rectangle. But we also apply the comma advantageously in such a case, for example, 1 as a,6+c, or AB,CD+EF; i.e., a multi- plied into 6+c, or AB into CD+EF; we speak about this soon, under vinculums. Formerly no sign of multiplication was considered neces- sary for, as stated above, commonly mere apposition sufficed. If, however, at any time a sign seems desirable use r^ rather than ><! , because the latter leads to ambiguity; accordingly, AB^\CD sig- nifies AB times CD. "Division is commonly marked by writing the divisor beneath its dividend, with a line of separation between them. Thus a divided by b is ordinarily indicated by r ; often, however, it is preferable to avoid this notation and to arrange the signs so that they are brought into one and the same line; this may be done by the interposition of two points; thus a:b signifies a divided by b. If a:b in turn is to be divided by c, we may write a : 6, : c, or (a : b) : c. However, this should be ex- pressed more simply in another way, namely, a : (be) or a : be, for the division cannot always be actually carried out, but can be only indicated, and then it becomes necessary to mark the delayed process of the operation by commas or parentheses Exponents are frequently inclosed by lines in this manner [a] (AB-\-BC), which means the cube of the line AB+BC . . . . ; the exponents of a l + n may also be advantageously written between the lines, so that the literal exponents will not be confounded with other letters; thus it- may be written \l+n\ a From ^(a 3 ) or j/ B (a 3 ) arises a . . . . ; but 1^2 or i/E 2 means the cube root of the same number, and $/% or V 02 signifies the extraction of a root of the indeterminate order e "For aggregation it is customary to resort to the drawing of 1 A similar use of the comma to separate factors and at the same time express aggregation occurs earlier in He*rigone (see 189). INDIVIDUAL WRITERS 225 lines, but because lines drawn one above others often occupy too much space, and for other reasons, it is often more convenient to introduce commas and parentheses. Thus a, b+c is the same as a, b+c or a(6+c) ; and a+b, c+d is the same as a+b, c+d, or (a+b) (c+d), i.e., +a+b multiplied by c+d. And, similarly, vinculums are placed under vinculums. For example, a, bc+ef+g is expressed also thus, a(bc+e(f+g)), and a, bc+ef+g+hlm,n may be written also + (a(bc+e(f+g))+hlm)n. What relates to vinculums in multiplica- tion applies to vinculums in division. For example, *+_!_ J. - may be written in one line thus: and there is no difficulty in this, as long as we observe that whatever fills up a given parenthesis be taken as one quantity ..... The same is true of vinculums in the extraction of roots. Thus ^a 4 +l/e, f+g is the same as ]/(a 4 +i/(e(/+fir))) or \/(a 4 +^ / (e, f+g)). And - ., /f . , , f , ,,. N , for - -- - ...... one may write y(aa+oy(cc+da)): 9 e+ Again a = b signifies that a is equal to 6, and a=~b signifies that a is greater than 6, and a=-b that a is less than b. Also proportionality or analogia of quantities, i.e., the identity of ratio, may be represented; we may express it in the calculus by the sign of equality, for there is no need of a special sign. Thus, we may indi- cate that a is to b as I is to ra by a:b = l:m, i.e., r = . The sign for o iii continued proportion is -Jf , so that TT a, b, c, and d are continued pro- portionals. "There is adopted a sign for similitude; it is <*> ; also a sign for both similitude and equality, or a sign of congruence, & accordingly, DEF&PQR signifies that the two triangles are similar; but DEF&. PRQ marks their congruence. Hence, if three quantities have to one another the same ratio that three others have to one another, we may mark this by a sign of similitude, asa;6;ccoZ;m;n means that a is to b as I is to m, and a is to c as / is to n, and b is to c as m is to n ..... " In the second edition of the Miscellanea Berolinensia, of the year 226 A HISTORY OF MATHEMATICAL NOTATIONS 1749, the typographical work is less faulty than in the first edition of 1710; some slight errors are corrected, but otherwise no alterations are made, except that Harriot's signs for "greater than" and "less than" are adopted in 1749 in place of the two horizontal lines of un- equal length and thickness, given in 1710, as shown above. 199. Conclusions. In a letter to Collins, John Wallis refers to a change in algebraic notation that occurred in England during his lifetime : "It is true, that as in other things so in mathematics, fashions will daily alter, and that which Mr. Oughtred designed by great letters may be now by others be designed by small; but a mathemati- cian will, with the same ease and advantage, understand Ac, and a 3 or aaa." 1 This particular diversity is only a trifle as compared with what is shown in a general survey of algebra in Europe during the fifteenth, sixteenth, and seventeenth centuries. It is discouraging to behold the extreme slowness of the process of unification. In the latter part of the fifteenth century p and m became symbols for "plus" and "minus" in France ( 131) and Italy ( 134). In Ger- many the Greek cross and the dash were introduced ( 146). The two rival notations competed against each other on European territory for many years. The p and m never acquired a foothold in Germany. The German + and gradually penetrated different parts of Europe. It is found in Scheubel's Algebra ( 158), in Recorders Whetstone of Witte, and in the Algebra of Clavius. In Spain the German signs occur in a book of 1552 ( 204), only to be superseded by the p and m in later algebras of the sixteenth century. The struggle lasted about one hundred and thirty years, when the German signs won out every- where except in Spain. Organized effort, in a few years, could have ended this more than a century competition. If one takes a cross-section of the notations for radical expressions as they existed in algebra at the close of the sixteenth century, one finds four fundamental symbols for indicating roots, the letters B and Z, the radical sign j/ proper and the fractional exponent. The letters 8 and / were sometimes used as capitals and sometimes as small letters ( 135, 318-22). The student had to watch his step, for at times these letters were used to mark, not roots, but the unknown quantity r and, perhaps, also its powers ( 136). When & stood for "root," it became necessary to show whether the root of one term or of several terms was meant. There sprang up at least seven different symbols For the aggregation of terms affected by the ft, namely, one of Chuquet ( 130), one of Pacioli ( 135), two of Cardan ( 141), the round paren- 1 See Rigaud, op. dt. t Vol. II, p. 475. INDIVIDUAL WRITERS 227 thesis of Tartaglia ( 351), the upright and inverted letter L of Bombelli ( 144), and the r bin. and r trinomia of A. V. Roomen ( 343). There were at least five ways of marking the orders of the root, those of Chuquet ( 130), De la Roche ( 132), Pacioli ( 135), Ghaligai ( 139), and Cardan (Fig. 46). With A. M. Visconti 1 the signs R.ce cu. meant the "sixth root"; he used the multiplicative principle, while Pacioli used the additive one in the notation of radicals. Thus the letter # carried with it at least fifteen varieties of usage. In con- nection with the letter /, signifying latus or "root/' there were at least four ways of designating the orders of the roots and the aggregation of terms affected ( 291, 322). A unique line symbolism for roots of different orders occurs in the manuscripts of John Napier ( 323). The radical signs for cube and fourth root had quite different shapes as used by Rudolff ( 148, 326) and Stifel ( 153). Though clumsier than StifePs, the signs of Rudolff retained their place in some books for over a century ( 328). To designate the order of the roots, Stifel placed immediately after the radical sign the German abbrevia- tions of the words zensus, cubus, zensizensuSj sursolidus, etc. Stevin ( 163) made the important innovation of numeral indices. He placed them within a circle. Thus he marked cube root by a radical sign followed by the numeral 3 coraled in a circle. To mark the root of an aggregation of terms, Rudolff ( 148, 348) introduced the dot placed after the radical sign ; Stifel sometimes used two dots, one before the expression, the other after. Stevin ( 163, 343) and Digges ( 334, 343) had still different designations. Thus the radical sign carried with it seven somewhat different styles of representation. Stevin suggested also the possibility of fractional exponents ( 163), the fraction being placed inside a circle and before the radicand. Altogether there were at the close of the sixteenth century twenty- five or more varieties of symbols for the calculus of radicals with which the student had to be familiar, if he desired to survey the publications of his time. Lambert Lincoln Jackson makes the following historical observa- tions: "For a hundred years after the first printed arithmetic many writers began their works with the line-reckoning and the Roman numerals, and followed these by the Hindu arithmetic. The teaching of numeration was a formidable task, since the new notation was so unfamiliar to people generally/' 2 In another place (p. 205) Jackson 1 "Abbreviationes," Praclica numerorum, ct mcnsurarum (Brescia, 1581). 2 The Educational Significance of Sixteenth Century Arithmetic (New York, 1906), p. 37, 38. 228 A HISTORY OF MATHEMATICAL NOTATIONS states: "Any phase of the growth of mathematical notation is an interesting study, but the chief educational lesson to be derived is that notation always grows too slowly. Older and inferior forms possess remarkable longevity, and the newer and superior forms appear feeble and backward. We have noted the state of transition in the sixteenth century from the Roman to the Hindu system of characters, the intro- duction of the symbols of operation, +, , and the slow growth toward the decimal notation. The moral which this points for twentieth-century teachers is that they should not encourage history to repeat itself, but should assist in hastening new improvements." The historian Tropfke expresses himself as follows: "How often has the question been put, what further achievements the patriarchs of Greek mathematics would have recorded, had they been in posses- sion of our notation of numbers and symbols! Nothing stirs the his- torian as much as the contemplation of the gradual development of devices which the human mind has thought out, that he might ap- proach the truth, enthroned in inaccessible sublimity and in its fullness always hidden from earth. Slowly, only very slowly, have these de- vices become what they are to man today. Numberless strokes of the file were necessary, many a chink, appearing suddenly, had to be mended, before the mathematician had at hand the sharp tool with which he could make a successful attack upon the problems con- fronting him. The history of algebraic language and writing presents no uniform picture. An assemblage of conscious and unconscious innovations, it too stands subject to the great world-law regulating living things, the principle of selection. Practical innovations make themselves felt, unsuitable ones sink into oblivion after a time. The force of habit is the greatest opponent of progress. How obstinate was the struggle, before the decimal division met with acceptation, before the proportional device was displaced by the equation, before the Indian numerals, the literal coefficients of Vieta, could initiate a world mathematics." 1 Another phase is touched by Treutlein: "Nowhere more than in mathematics is intellectual content so intimately associated with the form in which it is presented, so that an improvement in the latter may well result in an improvement of the former. Particularly in arithmetic, a generalization and deepening of concept became pos- sible only after the form of presentation had been altered. The his- tory of our science supplies many examples in proof of this. If the Greeks had been in possession of our numeral notation, would their 1 Tropfke, GeschichtederElementar-Mathematik, Vol. II (Leipzig, 1921), p. 4, 5. ADDITION AND SUBTRACTION 229 mathematics not present a different appearance? Would the binomial theorem have been possible without the generalized notation of pow- ers? Indeed could the mathematics of the last three hundred years have assumed its degree of generality without Vieta's pervasive change of notation, without his introduction of general numbers? These instances, to which others from the history of modern mathe- matics could be added, show clearly the most intimate relation between substance and form." 1 B. SPECIAL SURVEY OF THE USE OF NOTATIONS SIGNS FOR ADDITION AND SUBTRACTION 200. Early symbols. According to Hilprecht, 2 the early Baby- lonians had an ideogram, which he transliterates LAL, to signify "minus." In the hieratic papyrus of Ahines and, more clearly in the hieroglyphic translation of it, a pair of legs walking forward is the sign of addition; away, the sign of subtraction. 3 In another Egyptian papyrus kept in the Museum of Fine Arts in Moscow, 4 a pair of legs walking forward has a different significance; there it means to square a number. Figure 99, translated, is as follows (reading the figure from right to left) : "5 added and J- [of this sum] taken away, 10 remains. Make - f V of this 10: the result is 1, the remainder 9. | of it, namely, 6, added to it; the total is 15. of it is 5. When 5 is taken away, the remainder is 10." In the writing of unit fractions, juxtaposition meant addition, the unit fraction of greatest value being written first and the others in descending order of magnitude. While in Diophantus addition was expressed merely by juxtaposi- tion ( 102), a sporadic use of a slanting line / for addition, also a semi-elliptical curve 7 for subtraction, and a combination of the two 1 Treutlein, "Die deutsche Coss," Abhandlungen z. Geschichte der Mathematik, Vol. II (Leipzig, 1879), p. 27, 28. 2 H. V. Hilprecht, Babylonian Expedition: Mathematical etc. Tablets (Phila- delphia, 1906), p. 23. 3 A. Eisenlohr, op. cit. (2d ed.), p. 46 (No. 28), 47, 237. See also the improved edition of the Ahmes papyrus, The Rhind Mathematical Papyrus , by T. Eric Feet (London, 1923), Plate J, No. 28; also p. 63. 4 Peet, op. cit., p. 20, 135: Ancient Egypt (1917), p. 101. 230 A HISTORY OF MATHEMATICAL NOTATIONS P for the total result has been detected in Greek papyri. 1 Diophantus' sign for subtraction is well known ( 103). The Hindus had no mark for addition ( 106) except that, in the Bakhshali Arithmetic, yu is used for this purpose ( 109). The Hindus distinguished negative quantities by a dot ( 106, 108), but the Bakhshali Arithmetic uses the sign + for subtraction ( 109). The Arab al-Qalasadi in the fif- teenth century indicated addition by juxtaposition and had a special sign for subtraction ( 124). The Frenchman Chuquet (1484), the Italian Pacioli (1494), and the sixteenth-century mathematicians in Italy used p or p: for plus and m or m: for "minus" ( 129, 134). \ * FIG. 99. From the hieroglyphic translation of the Ahmes papyrus, Problem 28, showing a pair of legs walking forward, to indicate addition, and legs walking away, to indicate subtraction. (Taken from T. E. Peet, The Rhind Mathematical Papyrus, Plate J, No. 28.) 201. Origin and meanings of the signs + and . The modern algebraic signs + and came into use in Germany during the last twenty years of the fifteenth century. They are first found in manu- scripts. In the Dresden Library there is a volume of manuscripts, C. 80. One of these manuscripts is an algebra in German, written in the year 148 1, 2 in which the minus sign makes its first appearance in 1 H. Brugsch, Numerorum apud veteres Aegyptios demoticorum doctrina. Ex papyris (Berlin, 1849), p. 31; see also G. Friedlein, Zahlzeichen und das elementare Rechnen (Erlangen, 1869), p. 19 and Plate I. 2 E. Wappler, Abhandlungen zur Geschichte der Mathematik, Vol. IX (1899), p. 539, n. 2; Wappler, Zur Geschichte der deutschen Algebra im 15. Jahrhundert, Zwick- auer Gymnasialprogramm von 1887, p. 11-30 (quoted by Cantor, op. cit. t Vol. II [2d ed., 1900], p. 243, and by Tropfke, op. cit. t Vol. II [2d ed., 1921], p. 13). ADDITION AND SUBTRACTION 231 algebra (Fig. 100); it is called minnes. Sometimes the is placed after the term affected. In one case 4 is designated "4 das ist ." Addition is expressed by the word vnd. In a Latin manuscript in the same collection of manuscripts, C. 80, in the Dresden Library, appear both symbols + and as signs of operation (Fig. 101), but in some rare cases the + takes the place of et where the word does not mean addition but the general "and." 1 Repeatedly, however, is the word et used for addition. It is of no little interest that J. Widman, who first used 1. llteete* Minuaieiclien. the + and - in print, studied D*esd. C. 80. Deutsche Algebra, fol. 8G8' xl. J. ' X xl ( Um 1486 ) these two manuscripts in the manuscript volume C. 80 of t*\ &f zz the Dresden Library and, in fact, annotated them. One of the Dresden Library and, in his marginal notes is shown in ^ Fl - 1 '^ A ' m s l gl ? in a r -n- fno wj i 4. i MS c - 80 > Dresden Library. (Taken Figure 102. Widman lectured ^ j Tro ' pfke? op ^ VoL Y[1921], at the University of Leipzig, p< 14,) and a manuscript of notes taken in 1486 by a pupil is preserved in the Leipzig Library (Codex Lips. 1470) . 2 These notes show a marked resemblance to the two Dresden manuscripts. The view that our + sign descended from one of the florescent forms for et in Latin manuscripts finds further support from works on 2. iltestes Pluazeichen. Dread. C. 80. Lat Algebra, foL 350' 4 - Dread - C - 8() - (urn 1486) Lateinische Algebra, fol. 852* * + 2s* 10-* FIG. 101. Plus and minus signs in a Latin MS, C. 80, Dresden Library. (Taken from Tropfke, op. cit., Vol. II [2d ed., 1921], p. 14.) paleography. J. L. Walther 3 enumerates one hundred and two differ- ent abbreviations found in Latin manuscripts for the word et; one of these, from a manuscript dated 1417, looks very much like the modern 1 Wappler, Programm (1887), p. 13, 15. 2 Wappler, Zeitschrift Math. u. Physik, Vol. XLV (Hist. lit. Abt, 1900), p. 7-9. 8 Lexicon diplomaticvm abbreviationes syllabarvm et vocvm in diplomatibvs et codidbvs a secvlo VIII. ad XVI ..... Studio Joannis Lvdolfi VValtheri .... (Ulmae, 1756), p. 456-59. 232 A HISTORY OF MATHEMATICAL NOTATIONS + . The downward stroke is not quite at right angles to the horizontal stroke, thus -V. Concerning the origin of the minus sign ( ), we limit ourselves to the quotation of a recent summary of different hypotheses: "One knows nothing certain of the origin of the sign ; perhaps it is a simple bar used by merchants to separate the indication of the tare, for a long time called minus, from that of the total weight of merchan- dise; according to L. Rodet (Actes Zusatz von WIDMANN. Soc. philol Alen$on, Vol. VIII [1879], 5. Dresd^C. so, fol. 349' p 105 ) this sign was derived from an Egyptian hieratic sign. One has also _3 *"C 380 x sought the origin of our sign in the (44^ (T-i& 144- e x sign employed by Heron and Dio- phantus and which changed to T be- FIG. 102. Widmari's margin- * ., . ^i A -n i id note to MS C. 80, Dresden fore li became ~' Others stl11 havc Library. (Taken fromTropfke.) advanced the view that the sign - has its origin in the 6/3e\6s of the Alex- andrian grammarians. None of these hypotheses is supported by plausible proof." 1 202. The sign + first occurs in print in Widman's book in the question: "Als in diese exepel 16 elln pro 9 fl -J vn | + JL eynss fl wy kume 36 elln machss alsso Addir -J- vn -\ vn J zu samen kumpt -jj I- eynss fl Nu secz vn machss nach der regl vn kume 22 fl 8 L eynsz fl dz ist gerad 3 hlr in gold." 2 In translation: "Thus in this example, 16 ells [are bought] for 9 florins [and] J- and + of a florin, what will 36 ells cost? Proceed thus: Add and \ and | obtaining fj of a florin. Now put down and proceed according to the rule and there results 22 florin, and / of a florin which is exactly 3 heller in gold." The + in this passage stands for "and." Glaisher considers this + a mis- print for vn (the contraction for vnnd, our "and"), but there are other places in Widman where + clearly means "and," as we shall see later. There is no need of considering this + a misprint. On the same leaf Widman gives a problem on figs. We quote from the 1498 edition (see also Fig. 54 from the 1526 edition) : 1 Encycloptdie des scien. math., Tome I, Vol. I (1904), p. 31, 32, n. 145. 2 Johann Widman, Behede vnd hubsche Rechenung auff alien Kauffmanschafft (Leipzig, 1489), unnumbered p. 87. Our quotation is taken from J. W. L. Glaish- er's article, "On the Early History of Signs -f- and and on the Early German Arithmeticians," Messenger of Mathematics, Vol. LI (1921-22), p. 6. Extracts from Widman are given by De Morgan, Transactions of the Cambridge Philosophical Society, Vol. XI, p. 205, and by Boncompagni, Bullelino, Vol. IX, p. 205. ADDITION AND SUBTRACTION 233 "Veygen. Itm Eyner Kaufft 13 lagel veygen vn nympt ye 1 ct pro 4 fl % ort Vnd wigt itliche lagel als dan hye nochuolget. vn ich wolt wissen was an der sum brecht 4+ 5 Wiltu dass 4 17 wyssen der 3+36 dess gleichn 419 Szo sum 3+44 mir die ct 3+22 Vnd Ib vn Czentner 3 11 Ib was ist 3+50 dz ist mi 9 416 dz secz besu 3+44 der vn wer 3+29 de 4539 3-12 lb(Sodu 3+9 die ct zcu Ib gemacht hast Vnnd das + das ist mer dar zu addirest) vnd 75 min 9 Nu solt du fur holcz abschlahn albeg fur eyn lagel 24 Ib vn dz ist 13 mol 24' vn macht 312 Ib dar zu addir dz dz ist 75 Ib vnnd werden 387 Die subtrahir vonn 4539 Vnnd pleybn 4152 Ib Nu sprich 100 Ib das ist 1 ct pro 4 fl | wie kummen 4152 Ib vnd kumen 171 fl 5 ss 4 hlr | Vn ist recht gemacht. m In free translation the problem reads: "Figs. Also, a person buys 13 barrels of figs and receives 1 centner for 4 florins and \ ort (4J flor- ins), and the weight of each barrel is as follows: 4 ct+5 Ib, 4 ct 17 Ib, 3 ct+36 Ib, 4 ct- 19 Ib, 3 ct+44 Ib, 3 ct+22 Ib, 3 ct- 11 Ib, 3 ct+50 Ib, 4 ct-16 Ib, 3 ct+44 Ib, 3 ct+29 Ib, 3 ct-12 Ib, 3 ct+9 Ib; and I would know what they cost. To know this or the like, sum the ct and Ib and what is , that is minus, set aside, and they become 4539 Ib (if you bring the centners to Ib and thereto add the +, that is more) and 75 minus. Now you must subtract for the wood 24 Ib for each barrel and 13 times 24 is 312 to which you add the , that is 75 Ib and it becomes 387 which subtract from 4,539 and there remains 4152 Ib. Now say 100 Ib that is 1 ct for 4J fl, what do 4152 Ib come to, and they come to 171 fl 5 ss 4| hlr which is right. " Similar problems are given by Widman, relating to pepper and soap. The examination of these passages has led to divergent opinions on the original significance of the + and . De Morgan suspected 1 The passage is quoted and discussed by Enestrom, Bibliotheca mathematica, Vol. IX (3d ser., 1908-9), p. 156, 157, 248; see also ibid., Vol. VIII, p. 199. 234 A HISTORY OF MATHEMATICAL NOTATIONS that they were warehouse marks, expressing excess or deficiency in weights of barrels of goods. 1 M. W. Drobisch, 2 who was the first to point out the occurrence of the signs + and in Widman, says that Widman uses them in passing, as if they were sufficiently known, merely remarking, "Was ist das ist minus vnd das + das ist mer." C. I. Gerhardt, 3 like De Morgan, says that the + and were de- rived from mercantile practice. But Widman assigned the two symbols other significations as well. In problems which he solved by false position the error has the + or sign prefixed. 4 The was used also to separate the terms of a proportion. In "11630-198 4610-78" it separates the first and second and the third and fourth terms. The "78" is the computed term, the fractional value of the fourth term being omitted in the earlier editions of Widman's arithmetic. The sign + occurs in the heading "Regula augmenti + decrement!" where it stands for the Latin et ("and"), and is not used there as a mathematical symbol. In another place Widman gives the example, "Itm eyner hat kaufft 6 eyer 2 ^ pro 4 ^+1 ey" ("Again, someone has bought 6 eggs 2 ^ for 4 ^ + 1 egg"), and asks for the cost of one egg. Here the is simply a dash separating the words for the goods from the price. From this and other quotations Glaisher concludes that Widman used + and "in all the ways in which they are used in algebra." But we have seen that Widman did not restrict the signs to that usage; the + was used for "and" when it did not mean addition; the was used to indicate separation. In other words, Widman does not re- strict the use of + and to the technical meanings that they have in algebra. 203. In an anonymous manuscript, 5 probably written about the time when Widman's arithmetic appeared, use is made of symbolism in the presentation of algebraic rules, in part as follows: "Conditiones circa + vel in additione + et +\ - . +\ addalur non sumendo respectwn quis numerus sit /facit > et / / superior. 1 De Morgan, op. cit., Vol. XI, p. 206. 2 De Joannis Widmanni .... compendia (Leipzig, 1840), p. 20 (quoted by Glaisher, op. cit., p. 9). *Geschichte der Mathematik in Deutschland (1877), p. 36: ". . . . dass diese Zeichen im kaufmilnnischen Verkehr ublich waren." 4 Glaisher, op. cit., p. 15. 5 Regidae Cosae vel Algebrae, a Latin manuscript, written perhaps about 1450, but "surely before 1510," in the Vienna Library. ADDITION AND SUBTRACTION 235 . f .. I + et \ simpliciter subtrahatur minor numerus a Siiuent { , , > , -i MX A in I et +/ majon et residue sua ascribatur nota, * and similarly for subtraction. This manuscript of thirty-three leaves is supposed to have been used by Henricus Grammateus (Heinrich Schreiber) in the preparation of his Rechenbuch of 1518 and by Chris- toff Rudolff in his Coss of 1525. Grammateus 2 in 1518 restricts his use of + and to technical algebra: "Vnd man braucht solche zaichen als + 1st vnnd, myn- nder" ("And one uses such signs as + [which] is 'and/ ~ 'less' "). See Figure 56 for the reproduction of this passage from the edition of 1535. The two signs came to be used freely in all German algebras, particularly those of Grammateus, Rudolff (1525), Stifel (1544), and in Riese's manuscript algebra (1524). In a text by Eysenhut 3 the + is used once in the addition of fractions; both + and are employed many times in the regula falsi explained at the end of the book. Arithmetics, more particularly commercial arithmetics, which did not present the algebraic method of solving problems, did not usually make use of the + and symbols. L. L. Jackson says: "Although the symbols + and were in existence in the fifteenth century, and appeared for the first time in print in Widrnan (1489), as shown in the illustration (p. 53), they do not appear in the arithmetics as signs of operation until the latter part of the sixteenth century. In fact, they did not pass from algebra to general use in arithmetic until the nine- teenth century. " 4 204. Spread of the + and symbols. In Italy the symbols p and m served as convenient abbreviations for "plus" and "minus" at the end of the fifteenth century and during the sixteenth. In 1608 the German Clavius, residing in Rome, used the + and in his algebra brought out in Rome (see Fig. 66). Camillo Gloriosi adopted them in his Ad theorema geometricum of 1613 and in his Exercitationes mathematicae, decas I (Naples, 1627) ( 196). The + and signs were used by B. Cavalieri (see Fig. 86) as if they were well known. The + 1 C. I. Gerhardt, "Zur Geschichte der Algebra in Deutschland," Monats- berichte der k. pr. Akademie d. Wissenschaften z. Berlin (1870), p. 147. 2 Henricus Grammateus, Ayn New Kunsllich Buech (Niirnberg: Widrnung, 1518; publication probably in 1521). See Glaisher, op. cit., p. 34. 3 Ein kunsllich rechenbuch auff Zyffern / Lini vnd Wdlschen Practica (Augs- burg, 1538). This reference is taken from Tropfke, op. cit., Vol. I (2d ed., 1921), p. 58. 4 The Educational Significance of Sixteenth Century Arithmetic (New York, 1906), p. 54. 236 A HISTORY OF MATHEMATICAL NOTATIONS and were used in England in 1557 by Robert Recorde (Fig. 71) and in Holland in 1637 by Gillis van der Hoecke (Fig. 60). In France and Spain the German + and , and the Italian p and m, came in sharp competition. The German Scheubel in 1551 brought out at Paris an algebra containing the + and (158); nevertheless, the p and m (or the capital letters P, M) were retained by Peletier (Figs. 80, 81), Buteo (Fig. 82), and Gosselin (Fig. 83). But the adoption of the Ger- man signs by Ramus and Vieta (Figs. 84, 85) brought final victory for them in France. The Portuguese P. Nunez ( 166) used in his algebra (published in the Spanish language) the Italian p and m. Before this, Marco Aurel, 1 a German residing in Spain, brought out an algebra at Valencia in 1552 which contained the + and and the symbols for powers and roots found in Christoff Rudolff ( 165). But ten years later the Spanish writer P6rez de Moya returned to the Italian sym- bolism 2 with its p and w, and the use of n., co., ce, cu, for powers and r, rr, rrr for roots. Moya explains: "These characters I am moved to adopt, because others are not to be had in the printing office." 3 Of English authors 4 we have found only one using the Italian signs for "plus" and "minus," namely, the physician and mystic, Robert Fludd, whose numerous writings were nearly all published on the Continent. Fludd uses -P and M for "plus" and "minus." The + and , and the p and m, were introduced in the latter part of the fifteenth century, about the same time. They competed with each other for more than a century, and p and m finally lost out in the early part of the seventeenth century. 205. Shapes of the plus sign. The plus sign, as found in print, has had three principal varieties of form: (1) the Greek cross +, as it is found in Widman (1489); (2) the Latin cross, T more frequently placed horizontally, |- or H ; (3) the form *J, or occasionally some form still more fanciful, like the eight-pointed Maltese cross *%*, or a cross having four rounded vases with tendrils drooping from their edges. The Greek cross, with the horizontal stroke sometimes a little 1 Libro primero de Arithmetica Algebratica .... por Marco Aurel, natural Alcman (Valencia, 1552). 2 J. Rey Pastor, Los mathemdticos cspaiioles del siglo XVI (Oviedo, 1913), p. 38. 8 "Estos characteres me ha parecido poner, porque no auia otros eri la im- prenta" (Ad theorema geometricvm, d nobilissimo viro propositum, Joannis Camilli Gloriosi responsum [Venctiis, 1613], p. 26). 4 See C. Henry, Revue archeologigue, N.S., Vol. XXXVII, p. 329, who quotes from Fludd, Utriusque cosmi .... Historia (Oppenheim, 1617). ADDITION AND SUBTRACTION 237 longer than the vertical one, was introduced by Widman and has been the prevailing form of plus sign ever since. It was the form com- monly used by Grammateus, Rudolff, Stifel, Recorde, Digges, Clavius, Dee, Harriot, Oughtred, Rahn, Descartes, and most writers since their time. 206. The Latin cross, placed in a horizontal position, thus )-, was used by Vieta 1 in 1591. The Latin cross was used by Romanus, 2 Hunt, 3 Hume, 4 Hdrigone, 6 Mengoli, 6 Huygens, 7 Ferrnat, 8 by writers in the Journal des Sgavans, 9 Dechales, 10 Rolle, 11 Lamy, 12 L'Hospital, 13 Swedenborg, 14 Pardies, 16 Kresa, 16 Belidor, 17 De Moivre, 18 and Michel- sen. 19 During the eighteenth century this form became less common and finally very rare. Sometimes the Latin cross receives special ornaments in the form of a heavy dot at the end of each of the three shorter arms, or in the form of two or three prongs at each short arm, as in H. Vitalis. 20 A very ostentatious twelve-pointed cross, in which each of the four equal 1 Vieta, In artem analyticam isagoge (Turonis, 1591). 2 Adriani Romani Canon triangvlorvm sphaericorum .... (Mocvntiae, 1609). 3 Nicolas Hunt, The Hand-Maid to Arithmetick (London, 1633), p. 130. 4 James Hume, Traite de Valgebre (Paris, 1635), p. 4. 6 P. Herigone, "Explicatis notarvm," Cvrsvs mathematicvs, Vol. I (Paris, 1634). 6 Petro Merigoli, Geometriae speciosae elementa (Bologna, 1659), p. 33. 7 Chrisliani Hvgenii Holorogivm oscillatorivm (Paris, 1673), p. 88. 8 P. de Ferniat, Diophanli Alexandrini Arithmeticorum libri sex (Toulouse, 1670), p. 30; see also Fermat, Varia opera (1679), p. 5. 9 Op. cit. (Amsterdam, 1680), p. 160; ibid. (1693), p. 3, and other places. 10 K. P. Claudii Francisci Milliet Dechales, Mundus mathcmaticus, Vol. I (Leyden, 1690), p. 577. 11 M. Rolle, Methode pour resoudre les egalitez de tons Us degreez (Paris, 1691), p. 15. 12 Bernard Lamy, Siemens des mathematiques (3d ed.; Amsterdam, 1692), p. 61. 13 L'JTospital, Ada eruditorum (1694), p. 194; ibid. (1695), p. 59; see also other places, for instance, ibid. (1711), SuppL, p. 40. 14 Kmanuel Swedenborg, Daedalus Hyperborens (Upsala, 1716), p. 5; reprinted in Kungliga Vetenskaps Societetens i Upsala Tvdhundr adrsminne (1910). 15 (Euvres du R. P. Pardies (Lyon, 1695), p. 103. 16 J. Kresa, Analysis speciosa trigonometriae sphericae (Prague, 1720), p. 57. 17 B. F. de Belidor, Nouveau cours de mathematique (Paris, 1725), p. 10. 18 A. de Moivre, Miscellanea analytica (London, 1730), p. 100. 19 J. A. C. Michelsen, Theorie der Gleichungen (Berlin, 1791). 20 "Algebra," Lexicon mathematicum authore Hieronymo Vitali (Rome, 1690). 238 A HISTORY OF MATHEMATICAL NOTATIONS arms has three prongs, is given by Carolo Renaldini. 1 In seventeenth- and eighteenth-century books it is not an uncommon occurrence to have two or three forms of plus signs in one and the same publication, or to find the Latin cross in an upright or horizontal position, accord- ing to the crowded condition of a particular line in which the symbol occurs. 207. The cross of the form *% was used in 1563 and earlier by the Spaniard De Hortega, 2 also by Klcbotius, 3 Romanus, 4 and Des- cartes. 5 It occurs not infrequently in the Ada eruditorum 6 of Leipzig, and sometimes in the Miscellanea Berolinensia. 7 It was sometimes used by Halley, 8 Weigel, 9 Swedenborg, 10 and Wolff. 11 Evidently this symbol had a wide geographical distribution, but it never threatened to assume supremacy over the less fanciful Greek cross. A somewhat simpler form, + , consists of a Greek cross with four uniformly heavy black arms, each terminating in a thin line drawn across it. It is found, for example, in a work of Hindenburg, 12 and renders the plus signs on a page unduly conspicuous. Occasionally plus signs are found which make a "loud" display on the printed page. Among these is the eight-pointed Maltese cross, 1 Car oli Renaldini Ars analytica mathematicvm (Florence, 1665), p. 80, and throughout the volume, while in the earlier edition (Anconnae, 1644) he uses both the heavy eross and dagger form. 2 Fray Jua dc Hortega, Tractado subtilissimo d' arismetica 7 geometria (Gra- nada, 1563), leaf 51. Also (Seville, 1552), leaf 42. 3 Guillaume Klebitius, Insvlae Melitensis, quam alias Maltam vocant, Historia, Quacstionib. aliquot Mathcmalicis rcddila incundior (Diest [Belgium], 1565). I arn indebted to Professor H. Bosnians for information relating to this book. 4 Adr. Romanus, "Problema," Ideae malhematicae pars prima (Antwerp, 1593). 5 Ilene Descartes, La geometric (1637), p. 325. This form of the plus sign is in- frequent in this publication; the ordinary form (-f ) prevails. 6 See, for instance, op. cit. (1682), p. 87; ibid. (1683), p. 204; ibid. (1691), p. 179; ibid. (1694), p. 195; ibid. (1697), p. 131; ibid. (1698), p. 307; ibid. (1713), p. 344. 7 Op. cit., p. 156. However, the Latin cross is used more frequently than the form now under consideration. But in Vol. 11 (1723), the latter form is prevalent. 8 K Halley, Philosophical Transactions, Vol. XVII (London, 1692-94), p. 963; ibid. (1700-1701), Vol. XXII, p. 625. 9 Erhardi Weigelii Philosophia Mathematica (Jena, 1693), p. 135. 10 E. Swedenborg, op. cit., p. 32. The Latin cross is more prevalent in this book. 11 Christian Wolff, Mathcmatisches Lexicon (Leipzig, 1716), p. 14. 12 Carl Friedrich Hindenburg, Injinitinomii dignitaium .... leges ac Formulae (Gottingen, 1779). ADDITION AND SUBTRACTION 239 of varying shape, found, for example, in James Gregory, 1 Corachan, 2 Wolff, 3 and Hindenburg. 4 Sometimes the ordinary Greek cross has the horizontal stroke very much heavier or wider than the vertical, as is seen, for instance, in Fortunatus. 5 A form for plus / occurs in Johan Albert. 6 208. Varieties of minus signs. One of the curiosities in the his- tory of mathematical notations is the fact that notwithstanding the extreme simplicity and convenience of the symbol to indicate sub- traction, a more complicated symbol of subtraction -s- should have been proposed and been able to maintain itself with a considerable group of writers, during a period of four hundred years. As already shown, the first appearance in print of the symbols + and for "plus" and "minus" is found in Widman's arithmetic. The sign is one of the very simplest conceivable; therefore it is surprising that a modification of it should ever have been suggested. Probably these printed signs have ancestors in handwritten docu- ments, but the line of descent is usually difficult to trace with cer- tainty ( 201). The following quotation suggests another clue: "In the west-gothic writing before the ninth century one finds, as also Paoli remarks, that a short line has a dot placed above it , to indi- cate m y in order to distinguish this mark from the simple line which signifies a contraction or the letter N. But from the ninth century down, this same wcst-gothic script always contains the dot over the line even when it is intended as a general mark." 7 In print the writer has found the sign for "minus" only once. It occurs in the 1535 edition of the Rechenbuchlin of Grammateus (Fig. 56). He says: "Vnd man brauchet solche zeichen als + ist mehr / vnd / minder." 8 Strange to say, this minus sign does not occur in the first edition (1518) of that book. The corresponding pas- sage of the earlier edition reads: "Vnd man braucht solche zaichen 1 Geometriae pars vniversalis (Padua, 1668), p. 20, 71, 105, 108. 2 Juan Bautista Corachan, Arithmetica demonstrada (Barcelona, 1719), p. 326. 3 Christian Wolff, Elementa matheseos universae, Tomus I (Halle, 1713), p. 252. 4 Op. cit. 6 P. F. Fortunatus, Elementa matheseos (Brixia, 1750), p. 7. 8 Johan Albert, New Rechenbuchlein auff der federn (Wittembcrg, 1541); taken from Glaisher, op. cit., p. 40, 61. 7 Adriano Cappelli, Lexicon abbreviaturam (Leipzig, 1901), p. xx. 8 Henricus Grammateus, Eyn new Kunstlich behend and gewiss Rechenbuchlin (1535; 1st ed., 1518). For a facsimile page of the 1535 edition, see D. E. Smith, Kara arithmetica (1908), p. 125. 240 A HISTORY OF MATHEMATICAL NOTATIONS als + ist vnnd / mynnder." Nor does Grammateus use ~ in other parts of the 1535 edition; in his mathematical operations the minus sign is always . The use of the dash and two dots, thus -T-, for "minus," has been found by Glaisher to have been used in 1525, in an arithmetic of Adam Riese, 1 who explains: "Sagenn sie der warheit zuuil so be- zeychenn sie mit dem zeychen + plus wu aber zu wenigk so beschreib sie mit dem zeychen -5- minus genant." 2 No reason is given for the change from to -5-. Nor did Riese use -5- to the exclusion of . He uses -f- in his algebra, Die Coss, of 1524, which he did not publish, but which was printed 3 in 1892, and also in his arithmetic, published in Leipzig in 1550. Apparently, he used more frequently than -f- . Probably the reason for using -*- to designate lay in the fact that was assigned more than one signification. In Widman's arithmetic was used for subtraction or "minus," also for separating terms in proportion, 4 and for connecting each amount of an article (wool, for instance) with the cost per pound ( 202). The symbol was also used as a rhetorical symbol or dash in the same manner as it is used at the present time. No doubt, the underlying motive in introducing -f- in place of was the avoidance of confusion. This explanation receives support from the German astronomer Regio- montanus, 5 who, in his correspondence with the court astronomer at Ferrara, Giovanni Bianchini, used as a sign of equality; and used for subtraction a different symbol, namely, ip (possibly a florescent form of m). With him 1 ip r e meant 1 x. Eleven years later, in 1546, Gall Splenlin, of Ulm, had published at Augsburg his Arithmetica kunstlicher Rechnung, in which he uses -5-, saying: "Bedeut das zaichen -f zuuil, und das *- zii wenig." 6 Riese and Splenlin are the only arithmetical authors preceding the middle of the sixteenth century whom Glaisher mentions as using -~ for sub- traction or "minus." 7 Caspar Thierfeldern, 8 in his Arithmetica 1 Rechenung auff der linihen vndfedern in zal, masz, vnd gewicht (Erfurt, 1525; 1st ed., 1522). 2 This quotation is taken from Glaisher, op. cit., p. 36. 3 See Bruno Berlet, Adam Riese (Leipzig, Frankfurt am Main, 1892). 4 Glaisher, op. cit. t p. 15. 6 M. Curtze, Abhandlungen zur Geschichte der mathematischen Wissenschaften, Vol. XII (1902), p. 234; Karpinski, Robert of Chester, etc., p. 37. 6 See Glaisher, op. cit. t p. 43. 7 Ibid., Vol. LI, p. 1-148. 8 See Jackson, op. cit., p. 55, 220. ADDITION AND SUBTRACTION 241 (Nuremberg, 1587), writes the equation (p. 110), "18 fl.-v-85 gr. gleich25fl.-r-232gr." With the beginning of the seventeenth century -f- for "minus" appears more frequently, but, as far as we have been able to ascertain only in German, Swiss, and Dutch books. A Dutch teacher, Jacob Vander Schuere, in his Arithmetica (Haarlem, 1600), defines + and , but lapses into using -4- in the solution of problems. A Swiss writer, Wilhelm Schey, 1 in 1600 and in 1602 uses both -r- and TT for "minus." He writes 9+9, 54-12, 6-f-28, where the first number sig- nifies the weight in centner and the second indicates the excess or deficiency of the respective "pounds." In another place Schey writes "9 fl. -rr 1 ort," which means "9 florins less 1 ort or quart." In 1601 Nicolaus Reymers, 2 an astronomer and mathematician, uses regularly f- for "minus" of subtraction; he writes "XXVIII XII X VI III I Igr. 65532+18 -v-30 -18 +12-5-8" for z 28 =65,532z 12 +18z 10 -30z 6 -18z 3 +12z-~8 . Peter Roth, of Niirnberg, uses 44- in writing 3 3x 2 26z. Johannes Faulhaber 4 at Ulm in Wiirttemberg used -5- frequently. With him the horizontal stroke was long and thin, the dots being very near to it. The year following, the symbol occurs in an arithmetic of Ludolf van Ceulen, 6 who says in one place: "Subtraheert |/7 van, \/13, rest 1/13, weynigher j/7, daerom stelt |/13 voren en j/7 achter, met een sulck teecken -*- tusschen beyde, vvelck teecmin beduyt, comt alsoo de begeerde rest j/13-r-j/7 ." However, in some parts of the book is used for subtraction. Albert Girard 6 mentions -5- as the symbol for "minus," but uses . Otto Wesellow 7 brought out a book in which 1 Arithmetica oder die Kunst zu rechnen (Basel, 1600-1602). We quote from D. E. Smith, op. cit., p. 427, and from Matthiius Sterner, Geschichte der Rechen- kunst (Munchen and Leipzig, 1891), p. 280, 291. 2 Nicolai Raimari Ursi Dithmarsi .... arithmetica analytica, vulgo Cosa, oder Algebra (zu Frankfurt an der Oder, 1601). We take this quotation from Gerhardt, Geschichte der Mathematik in Deutschland (1877), p. 85. 3 Arithmetica philosophica (1608). We quote from Treutlein, "Die deutsche Coss," Abhandlungen zur Geschichte der Mathematik, Vol. II (Leipzig, 1879), p. 28, 37, 103. 4 Numerus figuratus sive arithmetica analytica (Ulm, 1614), p. 11, 16. 6 De arithmetische en geometrische Fondamenten (1615), p. 52, 55, 56. 6 Invention nouvelle en Valgebre (Amsterdam, 1629), no paging. A facsimile edition appeared at Leiden in 1884, 7 Flores arithmetici (driidde vnde veerde deel; Bremen, 1617), p. 523. 242 A HISTORY OF MATHEMATICAL NOTATIONS + and -4- stand for "plus" and "minus," respectively. These signs are used by Follinus, 1 by Stampioen ( 508), by Daniel van Hovcke 2 who speaks of + as 'signifying "mer en -f- min.," and by Johann Ardiiser 3 in a geometry. It is interesting to observe that only thirteen years after the publication of Ardiiser's book, another Swiss, J. H. Rahn, finding, perhaps, that there existed two signs for subtraction, but none for division, proceeded to use -f- to designate division. This practice did not meet with adoption in Switzerland, but was seized upon with great avidity as the symbol for division in a far-off country, England. In 1670 -r was used for subtraction once by Huygens 4 in the Philosophical Transactions. Johann Hemelings 5 wrote -H- for "minus" and indicated, in an example, 14^ legions less 1250 men by "14 1/2 Legion -f- 1250 Mann." The symbol is used by Tobias Beutel, 6 who writes "81^1fi6561-f-162. R. + l. zenss" to represent our 81 1/6561 162#+ 2 . Kegel 7 explains how one can easily multiply by 41, by first multiplying by 6, then by 7, and finally sub- tracting the multiplicand; he writes "7-7-1." In a set of seventeenth- century examination questions used at Ntirnberg, reference is made to cossic operations involving quantities, "durch die Signa + und -f- connectirt." 8 The vitality of this redundant symbol of subtraction is shown by its continued existence during the eighteenth century. It was em- ployed by Paricius, 9 of Regensburg. Schlesser 10 takes ^ to represent 1 Hermannus Follinus, Algebra sive liber de rebus occultis (Coloniae, 1622), p. 113, 185. 2 Cyffer-Boeck .... (den tweeden Druck: Rotterdam, 1628), p. 129-33. 8 Geometriae theoricae et practicae. Oder von dem Feldmdssen (Zurich, 1646), fol. 75. 4 In a reply to Slusius, Philosophical Transactions, Vol. V (London, 1670), p. 6144. 5 Arithmetisch-Poetisch-u. Historisch-Erquick Stund (Hannover, 1660) ; Selbst- lehrendes Rechen-Buch .... durch Johannem Hemelingium (Frankfurt, 1678). Quoted from Hugo Grosse, Historische Rechenbucher des 16. and 17. Jahrhunderts (Leipzig, 1901), p. 99, 112. 6 Geometrische Gallerie (Leipzig, 1690), p. 46. 7 Johann Michael Kegel, New vermehrte arithmetica vulgaris et practica italica (Frankfurt am Main, 1696). We quote from Sterner, op. cit., p. 288. 8 Fr. Unger, Die Methodik der praktischen Arithmetik in historischer Ent- wickelung (Leipzig, 1888), p. 30. 9 Georg Heinrich Paricius, Praxis arithmetices (1706). We quote from Sterner, op. cit., p. 349. 10 Christian Schlesser, Arithmetisches Haupt-Schliissel .... Die Coss oder Algebra (Dresden and Leipzig, 1720). ADDITION AND SUBTRACTION 243 "minus oder weniger." It was employed in the Philosophical Transac- tions by the Dutch astronomer N. Cruquius; 1 -*- is found in Hubsch 2 and Crusius. 3 It was used very frequently as the symbol for subtrao tion and "minus" in the Maandelykse Mathematische Liefhebbery, Purmerende (1754-69). It is found in a Dutch arithmetic by Bartjens 4 which passed through many editions. The vitality of the symbol is dis- played still further by its regular appearance in a book by van Steyn, 6 who, however, uses in 1778. 6 Halcke states, "-f- of het teken van substractio minims of min.," 7 but uses nearly everywhere. Praal- der, of Utrecht, uses ordinarily the minus sign , but in one place 8 he introduces, for the sake of clearness, as he says, the use of -f- to mark the subtraction of complicated expressions. Thus, he writes " = -v-9^+2j/26." The -f- occurs in a Leipzig magazine, 9 in a Dresden work by Illing, 10 in a Berlin text by Schmeisser, 11 who uses it also in expressing arithmetical ratio, as in "2-f-6-$- 10." In a part of KliigePs 12 mathematical dictionary, published in 1831, it is stated that -s- is used as a symbol for division, "but in German arithmetics is employed also to designate subtraction." A later use of it for "minus," that we have noticed, is in a Norwegian arithmetic. 13 In fact, in Scandinavian 1 Op. tit., Vol. XXXIII (London, 1726), p. 5, 7. 2 J. G. G. Hiibsch, Arithmelica portensis (Leipzig, 1748). 8 David Arnold Crusius, Anweisung zur Rechen-Kunst (Halle, 1746), p. 54. 4 De vernieuwde Cyfferinge van Mr. Willvm Bartjens, .... vermeerderl ende verbetert, door Mr. Jan van Dam en van alle voorgaande Fauten gezuyvert door Klaas Bosch (Amsterdam, 1771), p. 174-77. 6 Gerard van Steyn, Liefhebbery der Reekenkonst (eerste deel; Amsterdam,' 1768), p. 3, 11, etc. 6 Ibid. (2 Deels, 2 Stuk, 1778), p. 16. 7 Mathematisch Zinnen-Confect .... door Paul Halcken .... Uyt het Hoog- duytsch vertaald . ... dor Jacob Oostwoud (Tweede Druk, Te Purmerende, 1768), p. 5. 8 Mathematische Voorstellen .... door .... Ludolf van Keulen .... door Laurens Praalder (Amsterdam, 1777), p. 137. 9 J. A. Kritter, Leipziger Magazin fur reine and angewandte Mathematik (hcrausgegeberi von J. Bernoulli und C. F. Hindenburg, 1788), p. 147-61. 10 Carl Christian Illing, Arithmetisches Handbuch fur Lehrer in den Schulen (Dresden, 1793), p. 11, 132. 11 Friedrich Schmeisser, Lehrbuch der reinen Mathesis (1. Theil, Berlin, 1817), p. 45, 201. 18 G. S. Ktugel, "Zeichen," Mathematisches Wdrterbuch. This article was writ- ten by J. A. Grunert. 18 G. C. Krogh, Regnebogfor Begyndere (Bergen, 1869), p. 15. 244 A HISTORY OF MATHEMATICAL NOTATIONS countries the sign -f- for "minus" is found occasionally in the twentieth century. For instance, in a Danish scientific publication of the year 1915, a chemist expresses a range of temperature in the words "fra+18 C. til ^ 18 C." 1 In 1921 Ernst W. Selmer 2 wrote "0,72 4- 0,65 = 0,07." The difference in the dates that have been given, and the distances between the places of publication, make it certain that this symbol -T- for "minus" had a much wider adoption in Germany, Switzerland, Holland, and Scandinavia than the number of our cita- tions would indicate. But its use seems to have been confined to Teutonic peoples. Several writers on mathematical history have incidentally called attention to one or two authors who used the symbol -r- for "minus," but none of the historians revealed even a suspicion that this symbol had an almost continuous history extending over four centuries. 209. Sometimes the minus sign appears broken up into two or three successive dashes or dots. In a book of 1610 and again of 1615, by Ludolph van Ceulen, 3 the minus sign occasionally takes the form . Richard Balam 4 uses three dots and says "3 7, 3 from 7"; he writes an arithmetical proportion in this manner: "2 -4 = 3 -5." Two or three dots are used in Ren6 Descartes' Geometric, in the writings of Marin Mersenne, 5 and in many other seventeenth- century books, also in the Journal des S$avans for the year 1686, printed in Amsterdam, where one finds (p. 482) "1 R 11" for 1 l/ il, and in volumes of that Journal printed in the early part of the eighteenth century. Herigone used ~ for "minus" ( 189), the being pre-empted for recta linea. From these observations it is evident that in the sixteenth and seventeenth centuries the forms of type for "minus" were not yet standardized. For this reason, several varieties were sometimes used on the same page. This study emphasizes the difficulty experienced even in ordinary 1 Johannes Boye Pctcrsen, Kgl. Danske Vidensk. Selskabs Skrifter, Nat. og. Math, Afd., 7. Raekke, Vol. XII (Kopenhagen, 1915), p. 330; sec also p. 221, 223, 226, 230, 238. 2 Skrifter utgit av Videnskapsselskapet i Kristiania (1921)," Historisk-filosofisk Klasse" (2. Bind; Kristiania, 1922), article by Ernst W. Selmer, p. 11; see also p. 28, 29, 39, 47. 3 Circvlo et adscriptis liber Omnia e vernaculo Latina fecit et annotationibus illustravit Willebrordus Snellius (Leyden, 1610), p. 128. 4 Algebra (London, 1653), p. 5. *Cogitata Physico-Mathematica (Paris, 1644), Praefatio generalis, "De Rationibus atque Proportionibus," p. xii, xiii. ADDITION AND SUBTRACTION 245 arithmetic and algebra in reaching a common world-language. Cen- turies slip past before any marked step toward uniformity is made. It appears, indeed, as if blind chance were an uncertain guide to lead us away from the Babel of languages. The only hope for rapid ap- proach of uniformity in mathematical symbolism lies in international co-operation through representative committees. 210. Symbols for "plus or minus" The to designate "plus or minus" was used by Albert Girard in his Tables 1 of 1626, but with the + interpolation of ou, thus "ou" The was employed by Oughtred in his Clavis mathematicae (1631), by Wallis, 2 by Jones 3 in his Synopsis, and by others. There was considerable experimentation on suitable notations for cases of simultaneous double signs. For example, in the third book of his Geometric, Descartes uses a dot where we would +PP write . Thus he writes the equation "+?/ 6 2p?/ 4 4ryyqqy>Q" and then comments on this: "Et pour les signes -f- ou que iay omis, s'il y a eu+p en la precedente Equation, il faut mettre en celle cy + 2p, ou s'il ya eu p, il faut mettre 2p; & au contraire s'il ya eu + r, il faut mettre 4r, ..." The symbolism which in the Mis- cellanea Berolinensia of 1710 is attributed to Leibniz is given in 198. A different notation is found in Isaac Newton's Universal Arith- metick: "I denoted the Signs of b and c as being indeterminate by the Note J-, which I use indifferently for + or , and its opposite T for the contrary." 4 These signs appear to be the + with half of the vertical stroke excised. William Jones, when discussing quadratic equations, says: "Therefore if V be put for the Sign of any Term, and A for the contrary, all Forms of Quadratics with their Solutions, will be reduced to this one. If xxVaxVb = Q then A^aaa Ab|*." 5 Later in the book (p. 189) Jones lets two horizontal dots represent any sign: "Suppose any Equation whatever, as x n . . ax 7 *"- 1 . . 6x n ~ 2 . . cx n ~* . . dx n ~ 4 , etc. . . A=0." A symbol 8 standing for was used in 1649 and again as late as 1695, by van Schooten 6 in his editions of Descartes' geometry, also 1 See Bibliotheca mathematica (3d ser., 1900), Vol. I, p. 66. 2 J. Wallis, Operum mathematicorum pars prima (Oxford, 1657), p. 250. 3 William Jones, Synopsis Palmariorum matheseos (London, 1706), p. 14. 4 Op. cit. (trans. Mr. Ralphson .... rev. by Mr. Cunn; London, 1728), p. 172; also ibid. (rev. by Mr. Cunn .... expl. by Theaker Wilder; London, 1769), p. 321. 6 Op. cit., p. 148. 6 Renati Descartes Geometria (Leyden, 1649), Appendix, p. 330; ibid. (Frank- furt am Main, 1695), p. 295, 444, 445. 246 A HISTORY OF MATHEMATICAL NOTATIONS by De Witt. 1 Wallis 2 wrote & for + or , and R for the contrary. The sign & was used in a restricted way, by James Bernoulli; 3 h6 says, "% significat + in pr. e in post, hypoth.," i.e., the symbol stood for + according to the first hypothesis, and for , according to the second hypothesis. He used this same symbol in his Ars con- jectandi (1713), page 264. Van Schooten wrote also # for + . It should be added that tf appears also in the older printed Greek books as a ligature or combination of two Greek letters, the omicron o and the upsilon v. The tf appears also as an astronomical symbol for the constellation Taurus. Da Cunha 4 introduced f and ', or ' and +', to mean that the upper signs shall be taken simultaneously in both or the lower signs shall be taken simultaneously in both. Oliver, Wait, and Jones 5 denoted positive or negative N by *N. 211. The symbol [a] was introduced by Kronecker 6 to represent or + 1 or 1, according as a was or + 1 or 1. The symbol "sgn" has been used by some recent writers, as, for instance, Peano, 7 Netto, 8 and Le Vavasseur, in a manner like this: "sgn A = +1" when A >0, "sgn A = l" when A<0. That is, "sgn A" means the "sign of A." Similarly, Kowalewski 9 denotes by "sgn $" +1 when $ is an even, and 1 when ^ is an odd, permutation. The symbol I/a 2 is sometimes taken in the sense 10 a, but in equa- tions involving i/ , the principal root +a is understood. 212. Certain other specialized uses of + and . The use of each of the signs + and in a double sense first, to signify addition and subtraction; second, to indicate that a number is positive and nega- tive has met with opposition from writers who disregarded the ad- vantages resulting from this double use, as seen in a ( 6)=a+6, 1 Johannis de Witt, Elementa Cvrvarvm Linearvm. Edita Opera Francisci a Schooten (Amsterdam, 1683), p. 305. 2 John Wallis, Treatise of Algebra (London, 1685), p. 210, 278. 3 Ada eruditorum (1701), p. 214. 4 J. A. da Cunha, Principles mathematicos (Lisbon, 1790), p. 126. B Treatise on Algebra (2d cd.; Ithaca, 1887), p. 45. 6 L. Kronecker, Werke, Vol. II (1897), p. 39. 7 G. Peano, Formulario mathematico, Vol. V (Turin, 1908), p. 94. 8 E. Netto and R. le Vavasseur, Encyclopedic des scien. math,, Tome I, Vol. II (1907), p. 184; see also A. Voss and J. Molk, ibid., Tome II, Vol. I (1912), p. 257, n. 77. 9 Gerhard Kowalew8ki t EinfuhrungindieDeterminantentheorie (Leipzig, 1909), p. 18. 10 See, for instance, Encyclopedic des scien. math., Tome II, Vol. I, p. 257, n. 77. ADDITION AND SUBTRACTION 247 and who aimed at extreme logical simplicity in expounding the ele- ments of algebra to young pupils. As a remedy, German writers proposed a number of new symbols which are set forth by Schmeisser as follows: "The use of the signs + and , not only for opposite magni- tudes .... but also for Addition and Subtraction, frequently pre- vents clearness in these matters, and has even given rise to errors. For that reason other signs have been proposed for the positive and negative. Wilkins (Die Lehre von d. entgegengesetzL Grossen etc., Brschw., 1800) puts down the positive without signs (+a = a) but places over the negative a dash, as in a=d. v. Winterfeld (An* fangsgr. d. Rechenk., 2te Aufl. 1809) proposes for positive the sign h or f, for negative H or ~|. As more scientific he considers the in- version of the letters and numerals, but unfortunately some of them as i y r, o, x, etc., and 0, 1, 8, etc., cannot be inverted, while others, by this process, give rise to other letters as &, d, p, q, etc. Better are the more recent proposals of Winterfeld, to use for processes of computa- tion the signs of the waxing and waning moon, namely for Addition ), for Subtraction (, for Multiplication }, for Division <(, but as he himself acknowledges, even these are not perfectly suitable Since in our day one does not yet, for love of correctness, abandon the things that are customary though faulty,, it is for the present probably better to stress the significance of the concepts of the positive and additive, and of the negative and subtractive, in instruction, by the retention of the usual signs, or, what is the same thing, to let the qualitative and quantitative significance of + and be brought out sharply. This procedure has the advantage moreover of more fully exercising the understanding. " l Wolfgang Bolyai 2 in 1832 draws a distinction between + and , and + and H- ; the latter meaning the (intrinsic) "positive" and "negative." If A signifies *-*B, then A signifies +#. 213. In more recent time other notations for positive and nega- tive numbers have been adopted by certain writers. Thus, Spitz 3 uses <-a and ->a for positive a and negative a, respectively. M^ray 4 prefers "a , "a; Pad6, 5 a p , a n ; Oliver, Wait, and Jones 6 employ an ele- 1 Friedrieh Schmeisser, op. cit. } p. 42, 43. 2 Tentamen (2d ed., T. L; Budapestini, 1897), p. xi. * C. Spitz, Lehrbuch der alg. Arilhmetik (Leipzig, 1874), p. 12. 4 Charles Mray, Lemons nouv. de V analyse infin., Vol. I (Paris, 1894), p. 11. 6 H. Fade*, Premieres legons d'algtbre &&m. (Paris, 1892), p. 5. 9 Op. cit., p. 5. 248 A HISTORY OF MATHEMATICAL NOTATIONS vated + or (as in +10, -10) as signs of "quality"; this practice has been followed in developing the fundamental operations in algebra by a considerable number of writers; for instance, by Fisher and Schwatt, 1 and by Slaught and Lennes. 2 In elementary algebra the special sym- bolisms which have been suggested to represent "positive number" or "negative number" have never met with wide adoption. Stolz and Gmeiner 3 write a, a, for positive a and negative a. The designa- tion .... ~3, ~2, -1, 0, +1, +2, +3, . . . . , occurs in Huntington's Continuum (1917), page 20. 214. A still different application of the sign + has been made in the theory of integral numbers, according to which Peano 4 lets a+ signify the integer immediately following a, so that a+ means the inte- ger (a+ 1). For the same purpose, Huntington 5 and Stolz and Gmeiner 6 place the + in the position of exponents, so that 5+ = 6. 215. Four unusual signs. The Englishman Philip Ronayne used in his Treatise of Algebra (London, 1727; 1st ed., 1717), page 4, two curious signs which he acknowledged were "not common," namely, the sign -e to denote that "some Quantity indefinitely Less than the Term that next precedes it, is to be added," and the sign e- that such a quantity is "to be subtracted," while the sign J> may mean "either -e or e- when it matters not which of them it is." We have not noticed these symbols in other texts. How the progress of science may suggest new r symbols in mathe- matics is illustrated by the composition of velocities as it occurs in Einstein's addition theorem. 7 Silberstein uses here # instead of + 216. Composition of ratios. A strange misapplication of the + sign is sometimes found in connection with the "composition" of NP AN ratios. If the ratios -^jr r and -^^ are multiplied together, the product 1 G. E. Fisher and I. J. Schwatt, Text-Book of Algebra (Philadelphia, 1898), p. 23. 2 H. E. Slaught and U. J. Lennes, High School Algebra (Boston, 1907), p. 48. 1 Otto Stolz und J. A. Gmeiner, Theoretische Arithmetik (2d ed. ; Lej'pzig, 1911), Vol. I, p. 116. 4 G. Peano, Arithmetices principia nova methodo exposita (Turin, 1889); "Sul concetto di numero," Rivista di matem., Vol. I, p. 91; Formulaire de mathe- matiques, Vol. II, 2 (Turin, 1898), p. 1. 8 E. V. Huntington, Transactions of the American Mathematical Society, Vol. VI (1905), p. 27. 6 Op. cit., Vol. I, p. 14. In the first edition Peano's notation was used. 7 C. E. Weatherburn, Advanced Vector Analysis (London, 1924), p. xvi. ADDITION AND SUBTRACTION 249 NP AN ?T\f'rW> accorc ling to an old phraseology, was "compounded" of the L/./V C-iV first two ratios. 1 Using the term "proportion" as synonymous with "ratio," the expression "composition of proportions" was also used. As the word "composition" suggests addition, a curious notation, using +, was at one time employed. For example, Isaac Barrow 2 de- NP AN noted the "compounded ratio" 77^7 'T^F in this manner, "A r PCA r + C./V C./V AN'CN." That is, the sign of addition was used in place of a sign of multiplication, and the dot signified ratio as in Oughtred. In another book 3 Barrow again multiplies equal ratios by equal ratios. In modern notation, the two equalities are (PL+QO):QO = 2BC:(BC-CP) and QO:BC = BC:(BC+CP) . Barrow writes the result of the multiplication thus: PL+QO.QO+QO-BC=2BC-BC-CP+BC-BC+CP . Here the + sign occurs four times, the first and fourth times as a symbol of ordinary addition, while the second and third times it occurs in the "addition of equal ratios" which really means the multi- plication of equal ratios. Barrow's final relation means, in modern notation, BC ^ _ QO " * BC BC-CP ' BC+CP ' Wallis, in his Treatise of Algebra (London, 1685), page 84, com- ments on this subject as follows: "But now because Euclide gives to this the name of Composition, which word is known many times to im- part an Addition; (as when we say the Line ABC is compounded of AB and BC;) some of our more ancient Writers have chanced to call it Addition of Proportions; and others, following them, have continued that form of speech, which abides in (in divers Writers) even to this day: And the Dissolution of this composition they call Subduction of Proportion. (Whereas that should rather have been called Multi- plication, and this Division.)" A similar procedure is found as late as 1824 in J. F. Lorenz' trans- 1 See Euclid, Elements, Book VI, Definition 5. Consult also T. L. Heath, The Thirteen Books of Euclid's "Elements," Vol. II (Cambridge, 1908), p. 132-35, 189, 190. 2 Lectiones opticae (1669), Lect. VIII, V, and other places. 1 Lectiones geometricae (1674), Lect. XI, Appendix I, V. 250 A HISTORY OF MATHEMATICAL NOTATIONS lation from the Greek of Euclid's Elements (ed. C. B. Mollweide; Halle, 1824), where on page 104 the Definition 5 of Book VI is given thus: "Of three or more magnitudes, A } B> C, D, which are so related to one another that the ratios of any two consecutive magnitudes A : B, B:C, C : D, are equal to one another, then the ratio of the first magnitude to the last is said to be composed of all these ratios so that :C) + (C:D)" An modern notation, ~ = ~ SIGNS OF MULTIPLICATION 217. Early symbols. In the early Babylonian tablets there is, according to Hilprecht, 1 an ideogram A-DU signifying "times" or multiplication. The process of multiplication or division was known to the Egyptians 2 as wshtp, "to incline the head"; it can hardly be regarded as being a mathematical symbol. Diophantus used no symbol for multiplication ( 102). In the Bakhshali manuscript multiplication is usually indicated by placing the numbers side by side ( 109). In some manuscripts of Bhaskara and his commentators a dot is placed between factors, but without any explanation ( 112). The more regular mark for product in Bhaskara is the abbreviation bha, from bhavita, placed after the factors ( 112). Stifel in his Deutsche Arithmetica (Nurnberg, 1545) used the capital letter M to designate multiplication, and D to designate division. These letters were again used for this purpose by S. Stevin 3 who expresses our Sxyz 2 thus: 3 M sec M ter @, where sec and ter mean the "second" and "third" unknown quantities. The M appears again in an anonymous manuscript of 1638 ex- plaining Descartes' Geometric of 1637, which was first printed in 1896 ; 4 also once in the Introduction to a book by Bartholinus. 5 Vieta indicated the product of A and B by writing "A in B" (Fig. 84). Mere juxtaposition signified multiplication in the Bakhs- hali tract, in some fifteenth-century manuscripts, and in printed algebras designating 62 or 5# 2 ; but 5J meant 5+|, not 5X$. 1 H. V. Hilprecht, Babylonian Expedition, Vol. XX, Part 1, Mathematical etc. Tablets (Philadelphia, 1906), p. 16, 23. 2 T. Eric Peet, The Rhind Mathematical Papyrus (London, 1923), p. 13. 3 (Euvres mathematiques (ed. Albert Girard; Leyden, 1634), Vol. I, p. 7. 4 Printed in (Euvres de Descartes (e"d. Adam et Tannery), Vol. X (Paris, 1908), p. 669, 670. 6 Er. Bartholinus, Renati dea Cartes Principia matheseos universalis (Leyden, 1651), p. 11. See J. Tropfke, op. tit., Vol. II (2d ed., 1921), p. 21, 22. MULTIPLICATION 251 218. Early uses of the St. Andrew's cross, but not as a symbol of multiplication of two numbers. It is well known that the St. Andrew's cross (X) occurs as the symbol for multiplication in W. Oughtred's Clavis mathematicae (1631), and also (in the form of the letter X) in an anonymous Appendix which appeared in E. Wright's 1618 edi- tion of John Napier's Descriptio. This Appendix is very probably from the pen of Oughtred. The question has arisen, Is this the earliest use of X to designate multiplication? It has been answered in the negative incorrectly so, we think, as we shall endeavor to show. In the Encyclopedic des sciences mathematiques, Tome I, Volume I (1904), page 40, note 158, we read concerning X, "One finds it be- tween factors of a product, placed one beneath the other, in the Com- mentary added by Oswald Schreckenfuchs to Ptolemy's Almagest, 1551. "* As will be shown more fully later, this is not a correct inter- pretation of the symbolism. Not two, but four numbers are involved, two in a line and two others immediately beneath, thus: 315172^ ,295448 395093/ M74715 The cross does not indicate the product of any two of these numbers, but each bar of the cross connects two numbers which are multiplied. One bar indicates the product of 315172 and 174715, the other bar the product of 395093 and 295448. Each bar is used as a symbol singly; the two bars are not considered here as one symbol. Another reference to the use of X before the time of Oughtred is made by E. Zirkel, 2 of Heidelberg, in a brief note in which he protests against attributing the "invention" of X to Oughtred; he states that it had a period of development of over one hundred years. Zirkel does 1 Clavdii Ptolemaei Pelusierisis Alexandrini Omnia quae extant Opera (Basileae, 1551), Lib. ii, "Annotationcs." 2 Emil Zirkel, Zeitschr. f. math. u. naturw. Vnterricht, Vol. LII (1921), p. 96. An article on the sign X , which we had not seen before the time of proofreading, when R. C. Archibald courteously sent it to us, is written by N. L. W. A. Grave- laar in Wiskundig Tijdschrift, Vol. VI (1909-10), p. 1-25. Gravclaar cites a few writers whom we do not mention. His claim that, before Oughtred, the sign X occurred as a sign of multiplication, must be rejected as not borne out by the facts. It is one thing to look upon X as two symbols, each indicating a separate opera- tion, and quite another thing to look upon X as only one symbol indicating only one operation. This remark applies even to the case in 229, where the four num- bers involved are conveniently placed at the four ends of the cross, and each stroke connects two numbers to be subtracted one from the other. 252 A HISTORY OF MATHEMATICAL NOTATIONS not make his position clear, but if he does not mean that X was used before Oughtred as a sign of multiplication, his protest is pointless. Our own studies have failed to bring to light a clear and conclusive case where, before Oughtred, X was used as a symbol of multiplica- tion. In medieval manuscripts and early printed books X was used as a mathematical sign, or a combination of signs, in eleven or more different ways, as follows: (1) in solutions of problems by the process of two false positions, (2) in solving problems in compound proportion involving integers, (3) in solving problems in simple proportion involving fractions, (4) in the addition and subtraction of fractions, (5) in the division of fractions, (6) in checking results of computation by the processes of casting out the 9's, 7's, or ITs, (7) as part of a group of lines drawn as guides in the multiplication of one integer by another, (8) in reducing radicals of different orders to radicals of the same order, (9) in computing on lines, to mark the line indicating "thousands," (10) to take the place of the multiplication table above 5 times 5, and (11) in dealing with amicable numbers. We shall briefly discuss each of these in order. 219. The process of two false positions. The use of X in this process is found in the Liber abbaci of Leonardo 1 of Pisa, written in 1202. We must begin by explaining Leonardo's use of a single line or bar. A line connecting two numbers indicates that the two numbers are to be multiplied together. In one place he solves the problem: If 100 rotuli are worth 40 libras, how many libras are 5 rotuli worth? On the margin of the sheet stands the following: The line connecting 40 and 5 indicates that the two numbers are to be multiplied together. Their product is divided by 100, but no symbolism is used to indicate the division, Leonardo uses single lines over a hundred times in the manner here indicated. In more compli- cated problems he uses two or more lines, but they do not necessarily 1 Leonardo of Pisa, Liber abbaci (1202) (ed. B. Boncompagni; Roma, 1857), Vol. I, p. 84. MULTIPLICATION 253 form crosses. In a problem involving five different denominations of money he gives the following diagram: 1 barcellon. turn. Ian. pisan. imp, 20 A2^ 13 ^3k 12 11 13 13 barcellon. /* turn. \. Ian. .X^pisan, "X. imp. 11 M2^ 23 Here the answer 20+ is obtained by taking the product of the connected numbers and dividing it by the product of the unconnected numbers. Leonardo uses a cross in solving, by double false position, the problem: If 100 rotuli cost 13 libras, find the cost of 1 rotulus. The answer is given in solidi and denarii, where 1 libra = 20 solidi, 1 solidus = 12 denarii. Leonardo assumes at random the tentative answers (the two false positions) of 3 solidi and 2 solidi. But 3 solidi would make this cost of 100 rotuli 15 Zi&ra, an error of +2 libras; 2 solidi would make the cost 10, an error of 3. By the underlying theory of two false positions, the errors in the answers (i.e., the errors z 3 and x 2 solidi) are proportional to the errors in the cost of 100 rotuli (i.e., +2 and 3 libras); this proportion yields x = 2 solidi and 7-J denarii. If the reader will follow out the numerical operations for determining our x he will understand the following arrangement of the work given by Leonardo (p. 319) : u Additum ex 13 multiplicationibus 4 9 soldi soldi 9 minus 32 Additum ex erroribus. Jt Observe that Leonardo very skilfully obtains the answer by multiply- ing each pair of numbers connected by lines, thereby obtaining the products 4 and 9, which are added in this case, and then dividing 13 by 5 (the sum of the errors). The cross occurring here is not one sym- bol, but two symbols. Each line singly indicates a multiplication. It would be a mistake to conclude that the cross is used here as a symbol expressing multiplication, i Ibid., Vol. I, p. 127. 254 A HISTORY OF MATHEMATICAL NOTATIONS The use of two lines crossing each other, in double or single false position, is found in many authors of later centuries. For example, it occurs in MS 14908 in the Munich Library, 1 written in the interval 1455-64; it is used by the German Widman, 2 the Italian Pacioli, 3 the Englishman Tonstall, 4 the Italian Sfortunati, 5 the Englishman Recorde, 6 the German Splenlin, 7 the Italians Ghaligai 8 and Benedetti, 9 the Spaniard Hortega, 10 the Frenchman Trenchant, 11 the Dutchman Gemma Frisius, 12 the German Clavius, 18 the Italian Tartaglia, 14 the Dutchman Sncll, 15 the Spaniard Zaragoza, 16 the Britishers Jcake 17 and 1 See M. Curtze, Zeitschrift f. Math. u. Physik, Vol. XL (Leipzig, 1895). Supplement, Abhandlungen z. Geschichte d. Mathematik, p. 41. 2 Johann Widman, Behede vnd hubsche Rechenung (Leipzig, 1489). We have used J. W. L. Glaisher's article in Messenger of Mathematics, Vol. LI (1922), p. 16. 8 L. Pacioli, Summa de arithmetica, geometria, etc. (1494). We have used the 1523 edition, printed at Toscolano, fol. 99*, 10O, 182. 4 C. Tonstall, De arte supputandi (1522). We have used the Strassburg edi- tion of 1544, p. 393. 6 Giovanni Sfortunati da Siena, Nvovo Ijvme. Libro di Arithmetica (1534), fol. 89-100. 8 R. Recorde, Grovnd of Artes (1543[?]). We have used an edition issued be- tween 1636 and 1646 (title-page missing), p. 374. 7 Gall Splenlin, Arithmetica kiinstlicher Rechnung (1645). We have used J. W. L. Glaisher's article in op. cit., Vol. LI (1922), p. 62. 8 Francesco Ghaligai, Pratica d' arithmetica (Nuovamente Rivista ... ; Firenze, 1552), fol. 76. 9 lo. Baptistae Benedicti Divcrsarvm specvlationvm mathematicarum, et physica- rum Liber (Turin, 1585), p. 105. 10 Juan de Hortega, Tractado subtilissimo de arismetica y de geometria (emenda- do por Longalo Busto, 1552), fol. 138, 2156. 11 Jan Trenchant, L'arithmetiqve (4th ed.; Lyon, 1578), p. 216. 12 Gemma Frisius, Arithmeticae Practicae methodvs facilis (iam recens ab ipso authore emcndata .... Parisiis, 1569), fol. 33. 13 Christophori Clavii Bambergensis, Opera mathematica (Mogvntiae, 1612), Tomus secundus; "Numeratio," p. 58. 14 L'arithmetique de Nicolas Tartaglia Brescian (traduit par Gvillavmo Gosselin de Caen ... Premier Partie; Paris, 1613), p. 105. 15 Willebrordi Snelli Doctrinae Triangvlorvm Canonicae liber qvatvor (Leyden, 1627), p. 36. 16 Arithmetica Vniversal ... avthor El M. R. P. Joseph Zaragoza (Valencia, 1669), p. 111. 17 Samuel Jeake, AOriSTIKHAOriA or Arithmetick (London, 1696; Preface 1674), p. 501. MULTIPLICATION 255 Wingate, 1 the Italian Guido Grandi, 2 the Frenchman Chalosse, 3 the Austrian Steinmeyer, 4 the Americans Adams 5 and Preston. 6 As a sample of a seventeenth-century procedure, we give Schott's solution 7 of o~~~o = 30. He tries z=24 and x =48. He obtains & u o errors 25 and 20. The work is arranged as follows: 24.X r48. Dividing 48X25-24X20 by 5 iuf V M gives z= 144. M. A M. B 25. 5. 20. 220. Compound proportion with integers. We begin again with Leonardo of Pisa (1202) 8 who gives the problem: If 5 horses eat 6 quarts of barley in 9 days, for how many days will 16 quarts feed 10 horses? His numbers are arranged thus: The answer is obtained by dividing 9X16X5 by the product of the remaining known numbers. Answer 12. Somewhat different applications of lines crossing each other arc given by Nicolas Chuquet 9 and Luca Pacioli 10 in dealing with numbers in continued proportion. 1 Mr. Wingate 1 s Arithmetick, enlarged by John Kersey (llth ed.), with supple- ment by George Shelley (London, 1704), p. 128. 2 Guido Grandi, Instituzioni di arithmelia pratica (Firenze, 1740), p. 104. 1 L'arithmetique par les fractions ... par M. Chalosse (Paris, 1747), p. 158. 4 Tirocinium Arithmeticum a P. Philippo Steinmeyer (Vienna and Freiburg, 1763), p. 475. 6 Daniel Adams, Scholar's Arithmetic (10th ed.; Keene, N.H., 1816), p. 199. 8 John Preston, Lancaster's Theory of Education (Albany, N.Y., 1817), p. 349. 7 G. Schott, Cursus mathematicus (Wtirzburg, 1661), p. 36. 8 Op. cit., p. 132. Nicolas Chuquet, Le Triparty en la Science des Nombres (1484), edited by A. Marre, in BuUettino Boncompagni, Vol. XIII (1880), p. 700; reprint (Roma, 1881), p. 115. 10 Luca Pacioli, op. cit., fol. 93a. 256 A HISTORY OF MATHEMATICAL NOTATIONS Chuquet finds two mean proportionals between 8 and 27 by the scheme " 8 27 3 9 12 18 " where 12 and 18 are the two mean proportionals sought; i.e., 8, 12, 18, 27 are in continued proportion. 221. Proportions involving fractions. Lines forming a cross (X), together with two horizontal parallel lines, were extensively applied to the solution of proportions involving fractions, and constituted a most clever device for obtaining the required answer mechanically. If it is the purpose of mathematics to resolve complicated problems by a minimum mental effort, then this device takes high rank. The very earliest arithmetic ever printed, namely, the anonymous booklet gotten out in 1478 at Treviso, 1 in Northern Italy, contains an interesting problem of two couriers starting from Rome and Venice, respectively, the Roman reaching Venice in 7 days, the Venetian arriving at Rome in 9 days. If Rome and Venice are 250 miles apart, in how many days did they meet, and how far did each travel before they met? They met in 3}| days. The computation of the distance traveled by the courier from Rome calls for the solution of the pro- portion which we write 7:250 = -f-jj- : x. The Treviso arithmetic gives the following arrangement: 112 .250 63 16 The connecting lines indicate what numbers shall be multiplied to- gether; namely, 1, 250, and 63, also 7, 1, and 16. The product of the latter namely, 112 is written above on the left. The author then finds 250X63 = 15,750 and divides this by 112, obtaining 140| miles. These guiding lines served as Ariadne threads through the maze of a proportion involving fractions. We proceed to show that this magical device was used again by Chuquet (1484), Widman (1489), and Pacioli (1494). Thus Chuquet 2 1 The Treviso arithmetic of 1478 is described and partly given in facsimile by Boncompagni in Atti dell'Accademia Pontificia de' nuovi Lincei, Tome XVI (1862- 63; Roma, 1863), see p. 568. 2 Chuquet, in Boncompagni, Bullettino, Vol. XIII, p. 636; reprint, p. (84). MULTIPLICATION 257 uses the cross in the problem to find two numbers in the ratio of f 3\ /2 to | and whose sum is 100. He writes - X ~~> multiplying 3 by 3, 4/ X 3 and 2 by 4, he obtains two numbers in the proper ratio. As their sum is only 17, he multiplies each by \^ and obtains 47 T V and 52}?. Johann Widman 1 solves the proportion 9 : ^ 8 8 - = V : x in this man- 9 v/ 53 89 ner: "Secz also - j( machss nach der Regel vnd klipt 8 fl. 1 /X 8 8 35s 9 heir -f$" It will be observed that the computer simply took the products of the numbers connected by lines. Thus 1X53X89 = 4,717 gives the numerator of the fourth term; 9X8X8 = 576 gives the denominator. The answer is 8 florins and a fraction. Such settings of numbers are found in Luca Pacioli, ? Ch. Rudolph, 3 G. Sfortunati, 4 0. Schreckenfuchs, 5 Hortega, 6 Tartaglia, 7 M. Stein- metz, 8 J. Trenchant, 9 Hermann Follinus, 10 J. Alsted, 11 P. Hrigone, 12 Chalosse, 13 J. Perez de Moya. 14 It is remarkable that in England neither Tonstall nor Recorde used this device. Recorde 15 and Leonard Digges 16 1 Johann Widman, op. cit.; see J. W. L. Glaisher, op. cit., p. 6. 2 Luca Pacioli, op. cit. (1523), fol. 18, 27, 54, 58, 59, 64. 3 Christoph Rudolph, Kumlliche Rechnung (1526). We have used one of the Augsburg editions, 1574 or 1588 (title-page missing), CVII. 4 Giovanni Sfortunati da Siena, Nvovo Lvme. Libro di Arithmetica (1534), fol. 37. 6 O. Schreckenfuchs, op. cit. (1551). 8 Juan de Hortega, op. cit. (1552), fol. 92<z. 7 N. Tartaglia, General Trattato di Nvmeri (la prima partc, 1556), fol. 1116, 117a. 8 Arithmeticae Praecepta . . . . M. Mavricio Steinmetz Gersbachio (Leipzig, 1568) (no paging). 9 J. Trenchant, op. cit., p. 142. 10 Hermann vs Follinvs, Algebra sive liber de rebvs occvltis (Cologne, 1622), p. 72. 11 Johannis-Henrici Alstedii Encyclopaedia (Hernborn, 1630), Lib. XIV, Cossae libri III, p. 822. 12 Pierre Herigone, Cvrsvs mathematici, Tomus VI (Paris, 1644), p. 320. 18 U Arithmetique par les fractions ... par M. Chalosse (Paris, 1747), p. 71. 14 Juan Perez de Moya, Arithmetica (Madrid, 1784), p. 141. This text reads the same as the edition that appeared in Salamanca in 1562. 16 Robert Recorde, op. cit., p. 175. 16 (Leonard Digges), A Geometrical Practical Treatise named Pantometria (London, 1591). 258 A HISTORY OF MATHEMATICAL NOTATIONS use a slightly different and less suggestive scheme, namely, the capital letter Z for proportions involving either integers or fractions. Thus, *) i (* 3 : 8 = 16 : x is given by Recorde in the form Q ^^ . This rather un- usual notation is found much later in the American Accomptant of Chauncey Lee (Lansinburgh, 1797, p. 223) who writes, "Cause Effect" 4.5 yds. and finds Q = 90 X 18 + 4.5 = 360 dollars. 222, Addition and subtraction of fractions. Perhaps even more popular than in the solution of proportion involving fractions was the use of guiding lines crossing each other in the addition and subtrac- tion of fractions. Chuquet 1 represents the addition of f and by the following scheme: "10 12 " 5 15- The lower horizontal line gives 3X5 = 15; we have also 2X5 = 10, 3 X4 = 12; hence the sum f = 1-&. The same line-process is found in Pacioli, 2 Rudolph, 3 Apianus. 4 In England, Tonstall and Recorde do not employ this intersecting line-system, but Edmund Wingate 5 avails himself of it, with only slight variations in the mode of using it. We find it also in Oronce Fine, 8 Feliciano, 7 Schreckenfuchs, 8 Hortega, 9 Baeza, 10 the Italian 1 Nicolas Chuquet, op. tit., Vol. XIII, p. 606; reprint p. (54). 2 Luca Pacioli, op. cit. (1523), fol. 51, 52, 53. 8 Christoph Rudolph, op. cit., under addition and subtraction of fractions. 4 Petrus Apianus, Kauffmansz Rechnung (Ingolstadt, 1527). 6 E. Wingate, op. cit. (1704), p. 152. 6 Orontii Find Delphinatis, liberalivm Disdplinarvm professoris Regii Proto- mathesis: Opus varium (Paris, 1532), fol. 46. 7 Francesco Feliciano, Libro de arithmetica e geometria (1550). 8 O. Schreckenfuchs, op. cit., "Annot.," fol. 256. Hortega, op. cit. (1552), fol. 55a, 636. 10 Nvmerandi doctrina, authore Lodoico Baeza (Paris, 1556), fol. 386. MULTIPLICATION 259 translation of Fine's works, 1 Gemma Frisius, 3 Eygaguirre, 3 Clavius, 4 the French translation of Tartaglia, 5 Follinus, 6 Girard, 7 Hainlin, 8 Caramuel, 9 Jeake, 10 Corachan, 11 Chalosse, 12 De Moya, 13 and in slightly modified form in Crusoe. 14 223. Division of fractions. Less frequent than in the preceding processes is the use of lines in the multiplication or division of frac- tions, which called for only one of the two steps taken in solving a proportion involving fractions. Pietro Borgi (1488) 16 divides f by | 15 " thus: - x\ - In dividing \ by , Pacioli 16 writes 16 "2 3" and obtains | or 1|. Petrus Apianus (1527) uses the X in division, Juan de Hortega (1552) 17 divides f by |, according to the following scheme: , 9 1 Opere di Orontio Fineo del Definato. ... Tradotte da Cosimo Bartoli (Venice, 1587), fol. 31. 2 Arithmeticae Practicae methodvs facilis, per Gemmam Frisium ... iam recens ab ipso authore emendata ... (Paris, 1569), fol. 20. 3 Sebastian Fernandez Eycaguirre, Libra de Arithmetica (Brussels, 1608), p. 38. 4 Chr. Clavius, Opera omnia, Tom. I (1611), Euclid, p. 383. 6 L' Arithmetique de Nicolas Tartaglia Brescian, traduit ... par Gvillavmo Gossclin de Caen (Paris, 1613), p. 37. 6 Algebra sive Liber de Rebvs Occvltis, ... Hermann vs Follinvs (Cologne, 1622), p. 40. 7 Albert Girard, Invention Nouvelle en VAlgebre (Amsterdam, 1629). 8 Johan. Jacob Hainlin, Synopsis malhematica (Tubingen, 1653), p. 32. 8 Joannis Caramvelis Malhesis Biceps Veins et Nova (Companiae, 1670), p. 20. 10 Samuel Jeake, op. cit., p. 51. 11 Juan Bautista Corachan, Arithmetica demonstrada (Barcelona, 1719), p. 87. w L* Arithmetique par les fractions ... par M. Chalosse (Paris, 17^7), p. 8. 18 J. P. de Moya, op. cit. (1784), p. 103. 14 George E. Crusoe, Y Mathematics? ("Why Mathematics?") (Pittsburgh, Pa., 1921), p. 21. 15 Pietro Borgi, Arithmetica (Venice, 1488), fol. 33#. 18 L. Pacioli, op. cit. (1523), fol. 54a. 17 Juan de Hortega, op. cit. (1552), fol. 66a. 260 A HISTORY OF MATHEMATICAL NOTATIONS We find this use of X in division in Sfortunati, 1 Blundeville, 2 Steinmetz, 3 Ludolf van Ceulen, 4 De Graaf, 5 Samuel Jeake,* and J. Perez de Moya. 7 De la Chapelle, in his list of symbols, 8 introduces X as a regular sign of division, divisG par, and x as a regular sign of multiplication, multipli6 par. He employs the latter regularly in multiplication, but he uses the former only in the division of fractions, and he explains that in |Xf = $f, "le sautoir X montre que 4 doit multiplier 6 & que 3 doit multiplier 7," thus really looking upon X as two symbols, one placed upon the other. 224. In the multiplication of fractions Apianus 9 in 1527 uses the 1 o parallel horizontal lines, thus, ~ = . Likewise, Michael Stifel 10 uses A O two horizontal lines to indicate the steps. He says: "Multiplica numeratores inter se, et proveniet numerator productac summae. Multiplica etiam denominatores inter se, et proveniet denominator productae summae." 225. Casting out the 9's, 7's, or ll's. Checking results by casting out the 9's was far more common in old arithmetics than by casting out the 7's or 1 1's. Two intersecting lines afforded a convenient group- ing of the four results of an operation. Sometimes the lines appear in the form X, at other times in the form + Luca Pacioli 11 divides 97535399 by 9876, and obtains the quotient 9876 and remainder 23. Casting out the 7's (i.e., dividing a number by 7 and noting the residue), he obtains for 9876 the residue 6, for 97535399 the residue 3, "62 " for 23 the residue 2. He arranges these residues thus: >, Q - . u o Observe that multiplying the residues of the divisor and quotient, 6 times 6 = 36, one obtains 1 as the residue of 36. Moreover, 32 is also 1. This completes the check. 1 Giovanni Sfortvnati da Siena, Nvovo Lvme. Libra di Arithmetica (1534), fol. 26. 2 Mr. Blundevil. His Exercises conlayning eight Treatises (London, 1636), p. 29. 3 M. Mavricio Steinmetz Gersbachio, Arithmetical praecepta (1568) (no paging). 4 Ludolf van Ceulen, De arithm. (title-page gone) (1615), p. 13. 5 Abraham de Graaf, De Geheele Mathesis of Wiskonst (Amsterdam, 1694), p. 14. 9 Samuel Jeake, op. cU. t p. 58. 7 Juan Perez de Moya, op. cit., p. 117. 8 De la Chapelle, Institutions de geometric (4th eU; Paris, 1765), Vol. I, p. 44, 118, 185. * Petrus Apianus, op. cit. (1527). 10 M. Stifel, Arithmetica integra (Nuremberg, 1544), fol. 6. 11 Luca Pacioli, op. cit. (1523), fol. 35. MULTIPLICATION 261 Nicolas Tartaglia 1 checks, by casting out the 7's, the division 912345 + 1987 = 459 and remainder 312. Casting the 7's out of 912345 gives 0, out of 1987 gives 6, " 4'4 " out of 459 gives 4, out of 312 gives 4. Tartaglia writes down -j- . OiLF Here 4 times 6 = 24 yields the residue 3; minus 4, or better 7 minus 4, yields 3 also. The result "checks." Would it be reasonable to infer that the two perpendicular lines + signified multiplication? We answer "No," for, in the first place, the authors do not state that they attached this meaning to the symbols and, in the second place, such a specialized interpretation does not apply to the other two residues in each example, which are to be subtracted one from the other. The more general interpretation, that the lines are used merely for the convenient grouping of the four resi- dues, fits the case exactly. Rudolph 2 checks the multiplication 5678 times 65 = 369070 by casting out the 9's (i.e., dividing the sum of the digits by 9 and noting the residue); he finds the residue for the product to be 7, for the factors to be 2 and 8. He writes down Here 8 times 2 = 16, yielding the residue 7, written above. This residue is the same as the residue of the product; hence the check is complete. It has been argued that in cases like this Rudolph used X to indicate multiplication. This interpretation does not apply to other cases found in Rudolph's book (like the one which follows) and is wholly indefensible. We have previously seen that Rudolph used X in the addition and subtraction of fractions. Rudolph checks the proportion 9: 11 =48: x, where x = 58|, by casting out the 7's, 9's, and iTs as follows: "(7) (9) (II)' 1 \6/ vO, 0)/ Take the check by ITs (i.e., division of a number by 11 and noting the residue). It is to be established that 9# = 48 times 11, or that 9 1 N. Tartaglia, op. cit. (1556), fol. 34. a Chr. Rudolph, Kunstliche Rechnung (Augsburg, 1574 or 1588 ed.) A VIII. 262 A HISTORY OF MATHEMATICAL NOTATIONS times 528 = 48 times 99. Begin by casting out the ll's of the factors 9 and 48; write down the residues 9 and 4. But the residues of 528 and 99 are both 0. Multiplying the residues 9 and 0, 4 and 0, we obtain in each case the product 0. This is shown in the figure. Note that here we do not take the product 9 times 4; hence X could not possibly in- dicate 9 times 4. The use of X in casting out the 9's is found also in Recorders Grovnd of Aries and in Clavius 1 who casts out the 9's and also the 7's. Hortega 2 follows the Italian practice of using lines +, instead of X, for the assignment of resting places for the four residues con- sidered. Hunt 3 uses the Latin cross |- . The regular X is used by Regius (who also casts out the 7's), 4 Lucas, 5 Metius, 6 Alsted, 7 York, 8 Dechales, 9 Ayres, 10 and Workman. 11 In the more recent centuries the use of a cross in the process of casting out the 9 ? s has been abandoned almost universally; we have found it given, however, in an English mathematical dictionary 12 of 1814 and in a twentieth -century Portuguese cyclopedia. 13 226. Multiplication of integers. In Pacioli the square of 37 is found mentally with the aid of lines indicating the digits to be multi- plied together, thus: < 1369 ^hr. Clavius, Opera omnia (1612), Tom. I (1611), "Numeratio," p. 11. 2 Juan de Hortega, op. tit., fol. 426. 3 Nicolas Hunt, Hand-Maid to Arithmetick (London 1633). 4 Hudalrich Regius, Vtrivsgve Arithmetices Epitome (Strasburg, 1536), fol. 57; ibid. (Freiburg-in-Breisgau, 1543), fol. 56. 6 Lossius Lucas, Arithmelices Erotemata Pverilia (Liirieburg, 1569), fol. 8. 6 Adriani Metii Alcmariani Arithmeticae libri dvo: Leyden, Arith. Liber I, P. 11. 7 Johann Heinrich Alsted, Methodus Admirandorum mathcmaticorum novem Libris (Tertia editio; Herbon, 1641), p. 32. <Tho. York, Practical Treatise of Arithmetick (London, 1687), p. 38. 9 R. P. Claudii Francisci Milliet Dechales Camberiensis, Mundus Mathe- maticus. Tomus Primus, Editio altera (Leyden, 1690), p. 369. 10 John Ayres, Arithmetick made Easie, by E. Hatton (London, 1730), p. 53. 11 Benjamin Workman, American Accountant (Philadelphia, 1789), p. 25. 12 Peter Barlow, Math. <fc Phil. Dictionary (London, 1814), art. "Multiplica- tion." 13 Encyclopedia Portugueza (Porto), art. "Nove." MULTIPLICATION 263 From the lower 7 two lines radiate, indicating 7 times 7, and 7 times 3, Similarly for the lower 3. We have here a cross as part of the line- complex. In squaring 456 a similar scheme is followed ; from each digit there radiate in this case three lines. The line-complex involves three vertical lines and three well-formed crosses X. The multiplication of 54 by 23 is explained in the manner of Pacioli by Mario Bettini 1 in 1642. There are cases on record where the vertical lines are omitted, either as deemed superfluous or as the result of an imperfection in the typesetting. Thus an Italian writer, Unicorno, 2 writes: "7 8" x 4368 It would be a rash procedure to claim that we have here a use of X to indicate the product of two numbers; these lines indicate the product of 6 and 70, and of 50 and 8; the lines are not to be taken as one symbol; they do not mean 78 times 56. The capital letter X is used by F. Ghaligai in a similar manner in his Algebra. The same re- marks apply to J. H. Alsted 3 who uses the X, but omits the vertical lines, in finding the square of 32. A procedure resembling that of Pacioli, but with the lines marked as arrows, is found in a recent text by G. E. Crusoe. 4 227. Reducing radicals to radicals of the same order. Michael Stifel 6 in 1544 writes: "Vt volo reducere \/z 5 et j/c 4 ad idem signum, sic stabit exemplum ad regulam 5 4 x V* V4. 1 Mario Bettino, Apiaria Vniversae philosophiae mathematicae (Bologna, 1642), "Apiarivm vndecimvm," p. 37. 2 S. Joseppo Vnicorno, De Varithmetica universale (Venetia, 1598), fol. 20. Quoted from C. le Paige, "Sur 1'origine de certains signes d'opfaation," Anncdes de la soctitt scientifiyue de Bruxelles (16th year, 1891-92), Part II, p. 82. 8 J. H. Alsted, Methodus Admirandorum Maihematicorum Novem libris e&- hibens universam mathesin (tertiam editio; Herbon, 1641), p. 70. 4 George E. Crusoe, op. ct/., p. 6. 6 Michael Stifel, Arithmetica integra (1544), fol. 114. 264 A HISTORY OF MATHEMATICAL NOTATIONS j/zc125 et j/2c16," Here j/5 and ^4 are reduced to radicals of the same order by the use of the cross X. The orders of the given radicals are two and three, respectively ; these orders suggest the cube of 5 or 125 and the square of 4, or 16. The answer is 1/125 and 1/16. Similar examples are given by Stifel in his edition of RudolfTs Coss y l Peletier, 2 and by De Billy. 3 228. To mark the place for "thousands." In old arithmetics explaining the computation upon lines (a modified abacus mode of computation), the line on which a dot signified "one thousand" was marked with a X. The plan is as follows: X - - 1000 500 - 100 50 - 50 5 - 1 This notation was widely used in Continental and English texts. 229. In place of multiplication table above 5X5. This old pro- cedure is graphically given in Recorders Grovnd of Artes (1543?). Re- quired to multiply 7 by 8. Write the 7 and 8 at the cross as shown here; next, 10 8 = 2, 10 7, = 3; write the 2 and 3 as shown: "8 2" 56 Then, 2X3 = 6, write the 6; 7-2 = 5, write the 5. The required product is 56. We find this process again in Oronce Fine, 4 Regius, 6 1 Michael Stifel, Die Coss Christoffs Rudolfs (Amsterdam, 1615), p. 136. (First edition, 1553.) 2 Jacobi Peletarii Cenomani, de occvlta parte nvmerorvm, qvam Algebram vacant, Libri duo (Paris, 1560), fol. 52. s Jacqves de Billy, Abregt des Preceptes d'Algebre (Reims, 1637), p. 22. See also the Nova Geometriae Clavis, authore P. Jacobo de Billy (Paris, 1643), p. 465. 4 Orontii Finei Delphinatis, liberalivm Disciplinarvm prefossoris Regii Proto- mathesis: Opus uarium (Paris, 1532), fol. 4b. 6 Hudalrich Regius, Vtrivsqve arithmetices Epitome (Strasburg, 1536), fol. 53; ibid. (Freiburg-in-Breisgau, 1543), fol. 56. MULTIPLICATION 265 Stifel, 1 Boissiere, 2 Lucas, 3 the Italian translation of Oronce Fine, 4 the French translation of Tartaglia, 5 Alsted, 8 Bettini. 7 The French edition of Tartaglia gives an interesting extension of this process, which is exhibited in the product of 996 and 998, as follows: 996 994 8 230. Amicable numbers. N. Chuquet 8 shows graphically that 220 and 284 are amicable numbers (each the sum of the factors of the other) thus: "220 /2S4" 110\/ 142 4 2 1 220 44 22 20 11 10 5 4 2 1 284 The old graphic aids to computation which we have described are interesting as indicating the emphasis that was placed by early arith- meticians upon devices that appealed to the eye and thereby con- tributed to economy of mental effort. 231. The St. Andrew 1 s cross used as a symbol of multiplication. As already pointed out, Oughtred was the first ( 181) to use X as the 1 Michael Stifel, Arithmetica integra (Nuremberg, 1544), fol. 3. 2 Claude de Boissiere, Daulphinois, UArtd'Arythmetique (Paris, 1554), fol. 156. 3 Lossius Lucas, Arithmetices Erotemata Pverilia (Liineburg, 1569), fol. 8. 4 Opere di Orontio Fineo. ... Tradotte da Cosimo Bartoli (Bologna, 1587), "Delia arismetica," libro primo, fol. 6, 7. 6 L'arithmetique de Nicolas Tartaglia ... traduit ... par Gvillavmo Gosselin de Caen. (Paris, 1613), p. 14. 6 Johannis-Henrici Alstedii Encyclopaedia (Herbon, 1630), Lib. XIV, p. 810. * Mario Bettino, Apiaria (Bologna, 1642), p. 30, 31. 8 N. Chuquet, op. tit., VoL XIII, p. 621; reprint, p. (69). 266 A HISTORY OF MATHEMATICAL NOTATIONS sign of multiplication of two numbers, as aXb (see also 186, 288). The cross appears in Oughtred's Clavis mathematicae of 1631 and, in the form of the letter X, in E. Wright's edition of Napier's Descriptio (1618). Oughtred used a small symbol X for multiplication (much smaller than the signs + and ). In this practice he was followed by some writers, for instance, by Joseph Moxon in his Mathematical Dictionary (London, 1701), p. 190. It seems that some objection had been made to the use of this sign X, for Wallis writes in a letter of September 8, 1668: "I do not understand why the sign of multi- plication X should more trouble the convenient placing of the frac- tions than the older signs + = >:: ."* It may be noted that Oughtred wrote the X small and placed it high, between the factors. This practice was followed strictly by Edward Wells. 2 On the other hand, in A. M. Legendre's famous textbook Geometric (1794) one finds (p. 121) a conspicuously large-sized symbol X> f r multiplication. The following combination of signs was suggested by Stringham: 3 Since X means "multiplied by," and / "divided by," the union of the two, viz., X/, means "multiplied or divided by." 232. Unsuccessful symbols for multiplication. In the seventeenth century a number of other designations of multiplication were pro- posed. Hrigone 4 used a rectangle to designate the product of two factors that were separated by a comma. Thus, "D5+4+3, 7~3: ~10,es38" meant (5+4+3)- (7-3)-10 = 38. Jones, in his Synopsis palmariorum (1706), page 252, uses the us, the Hebrew letter mem, to denote a rectangular area. A six-pointed star was used by Rahn arid, after him, by Brancker, in his translation of Rahn's Teutsche Algebra (1659). "The Sign of Multiplication is [-#] i.e., multiplied with." We encounter this use of -X- in the Philosophical Transactions. 5 Abraham de Graaf followed a practice, quite common among Dutch writers of the seventeenth and eighteenth centuries, of placing symbols on the right of an expression to signify direct operations (multiplication, involution), and placing the same symbols on the 1 S. P. Rigaud, Correspondence of Scientific Men of the Seventeenth Century (Oxford, 1841), Vol. II, p. 494. * Edward Wells, The Young Gentleman's Arithmetic and Geometry <2d ed.; London, 1723); "Arithmetic," p. 16, 41; "Geometry," p. 283, 291. 3 Irving Stringham, Uniplanar Algebra (San Francisco, 1893), p. xiii. 4 P. Herigone, Cursus mathematici (1644), Vol. VI, explicatio notarum. (First edition, 1642.) 5 Philosophical Transactions, Vol. XVII, (1692-94), p. 680. See also 194, 547. MULTIPLICATION 267 left of an expression to signify inverse operations. Thus, Graaf 1 multiplies x*+4 by 2J by using the following symbolism: , " xx tot 4 . " als f xx tot 9 In another place he uses this same device along with double commas, thus <t 7 r a+b , , cc , a+b , , ccd to represent (a+b)( cc) (d) = (a + b) ( ccd) . Occasionally the comma was employed to mark multiplication, as 23 23 112 in Herigone ( 189), F. Van Schooten, 2 who in 1657 gives '- ~ , Q *2 '2 i /I 1 Q r 3,0,0 I/ 11O,0 where all the commas signify "times/' as in Leibniz ( 197, 198, 547), in De Gua 3 who writes "3, 4, 5 , ... &c. n w "2," in Petrus Hor- rebowius 4 who lets "A t B" stand for A times B, in Abraham de Graaf 5 who uses one or two commas, as in "p 6,a" for (p 6)a. The German Htibsch 6 designated multiplication by /-, as in l/^. 233. The dot for multiplication. The dot was introduced as a symbol for multiplication by G. W. Leibniz. On July 29, 1098, ho wrote in a letter to John Bernoulli: "I do not like X as a symbol for multiplication, as it is easily confounded with a:; .... often I simply relate two quantities by an interposed dot and indicate multiplication by ZC - LM. Hence, in designating ratio I use not one point but two poiiits, which I use at the same time for division." It has been stated that the dot was used as a symbol for multiplication before Leibniz, that Thomas Harriot, in his Artis analyticae praxis (1631), used the dot in the expressions "aaa 3-W>a=+2-ccc." Similarly, in explain- ing cube root, Thomas Gibson 7 writes, in 1655, "3-66," "3 -6cc," but it 1 Abraham de Graaf, Algebra of Stelkonst (Amsterdam, 1672), p. 8. 2 Francisci ct Schooten. ... Exercitationum mathematicarum liber primus (Leyden, 1657), p. 89. 3 L'Abbe' de Gua, Histoire de I'academie r. d. sciences, annee 1741 (Paris, 1744), p. 81. ' 4 Pelri Horrebowii Operum mathematico-physicorum tomus primus (Havniae, 1740), p. 4. 5 Abraham de Graaf, op. cit. (1672), p. 87. J. G. G. Htibsch, Arithmetica Portensis (Leipzig, 1748). Taken from Wilder- muth's article, "Rechnen," in K. A. Schmid ; s Encyklopaedie des gesammten Er- ziehungs- und Unterrichtswesens (1885). 7 Tho. Gibson, Syntaxis mathematica (London, 1655), p. 36. 268 A HISTORY OF MATHEMATICAL NOTATIONS is doubtful whether either Harriot or Gibson meant these dots for multiplication. They are introduced without explanation. It is much more probable that these dots, which were placed after numerical coefficients, are survivals of the dots habitually used in old manu- scripts and in early printed books to separate or mark off numbers appearing in the running text. Leibniz proposed the dot after he had used other symbols for over thirty years. In his first mathematical publication, the De arte combinatoria 1 of 1666, he used a capital letter C placed in the position O for multiplication, and placed in the position O for division. We have seen that in 1698 he advocated the point. In 1710 the Leibnizian symbols 2 were explained in the publica- tion of the Berlin Academy ( 198); multiplication is designated by apposition, and by a dot or comma (punctum vel comma), as in 3,2 or a,b+c or AB,CD+EF. If at any time some additional symbol is de- sired, O is declared to be preferable to X . The general adoption of the dot for multiplication in Europe in the eighteenth century is due largely to Christian Wolf. It was thus used by L. Euler; it was used by James Stirling in Great Britain, where the Oughtredian X was very popular. 3 Whit worth 4 stipulates, "The full point is used for the sign of multiplication." 234. The St. Andrew's cross in notation for transjinite ordinal numbers. The notation o>X2, with the multiplicand on the left, was chosen by G. Cantor in the place of 2co (where w is the first transfinite ordinal number), because in the case of three ordinal transfinite numbers, a, (I, 7, the product a? a 7 is equal to of^ when oP is the multiplicand, but when a y is the multiplicand the product is a 7 ^. In transfinite ordinals, +7 is not equal SIGNS FOR DIVISION AND RATIO 235. Early symbols. Hilprecht 5 states that the Babylonians had an ideogram IGI-GAL for the expression of division. Aside from their fractional notation ( 104), the Greeks had no sign for division. Diophantus 6 separates the dividend from the divisor by the words Iv 1 G. W. Leibniz, Opera omnia, Vol. II (Geneva, 1768), p. 347. 2 Miscellanea Berolinensia (Berlin), Vol. I (1710), p. 156. ' See also 188, 287, 288; Vol. II, 541, 547. 4 W. A. Whitworth, Choice and Chance (Cambridge, 1886), p. 19. 6 H. V. Hilprecht, The Babylonian Expedition Mathematical, etc., Tablets from the Temple Library of Nippur (Philadelphia, 1906), p. 22. 6 Diophantus, Arithmetica (ed. P. Tannery; Leipzig, 1893), p. 286. See also G. H. F. Nesselmann, Algebra der Griechen (Berlin, 1842), p. 299. DIVISION AND RATIO 269 or noplov, as in the expression 6 V J \d\f/a ss ic6 /ioptou 6a/x d IJ3 f, which means (7z 2 24x)-f-(x 2 +12 7z). In the Bakhshali arithmetic ( 109) division is marked by the abbreviation bhd from bhdga, "part." The Hindus often simply wrote the divisor beneath the dividend. Similarly, they designated fractions by writing the denominator beneath the numerator ( 106, 109, 113). The Arabic author 1 al-Ha?$ar, who belongs to the twelfth century, mentions the use of a fractional line in giving the direction: " Write the denomina- tors below a [horizontal] line and over each of them the parts belonging to it; for example, if you are told to write three-fifths and a third of a 3 1 fifth, write thus, H~O" ^ n a second example, four-thirteenths and O o 4 3 three-elevenths of a thirteenth is written ^ ^. This is the first lo 11 appearance of the fractional line, known to us, unless indeed Leonardo of Pisa antedates al-Ha$sar. That the latter was influenced in this matter by Arabic authors is highly probable. In his Liber abbaci (1202) he uses the fractional line ( 122). Under the caption 2 "De diuisionibus integrorum numcrorum" he says: "Cum super quem- libet numerum quedam uirgula protracta fuerit, et super ipsam qui- libet alius numerus descriptus fuerit, superior numerus partem uel partes inferioris numeri affirmat; nam inferior denominatus, et su- perior denominans appellatur. Vt si super binarium protracta fuerit uirgula, et super ipsam unitas descripta sit ipsa unitas unam part-cm de duabus partibus unius integri affirmat, hoc est medietatem sic |." ("When above any number a line is drawn, and above that is written any other number, the superior number stands for the part or parts of the inferior number; the inferior is called the denominator, the superior the numerator. Thus, if above the two a line is drawn, and above that unity is written, this unity stands for one part of two parts of an integer, i.e., for a half, thus .") With Leonardo, an indi- cated division and a fraction stand in close relation. Leonardo writes 157 also . o"r"Yri> w ^ich means, as he explains, seven-tenths, and five- Zi U lU sixths of one-tenth, and one-half of one-sixth of one-tenth. 236. One or two lunar signs, as in 8)24 or 8)24(, which are often employed in performing long and short division, may be looked upon as symbolisms for division. The arrangement 8)24 is found in Stifel's 1 H. Sutcr, Bibliotheca mathcmatica (3d scr.), Vol. II (1901), p. 24. 2 // Liber abbaci di Leonardo Pisano (ed. B. Boncompagni; Roma, 1857), p. 23, 24. 270 A HISTORY OF MATHEMATICAL NOTATIONS Arithmetica Integra (1544) 1 , and in W. Oughtred's different editions of his Claris mathematicae. In Oughtred's Opuscula posthuma one finds also $]|[f, (182). Joseph Moxon 2 lets D)A+B-C signify our Perhaps the earliest to suggest a special symbol for division other than the fractional line, and the arrangement 5)15 in the process of dividing, was Michael Stifel 3 in his Deutsche Arithmetica (1545). By the side of the symbols + and he places the German capitals 2ft and 35, to signify multiplication and division, respectively. Strange to say, he did not carry out his own suggestion; neither he nor seem- ingly any of his German followers used the 50} and SD in arithmetic or algebraic manipulation. The letters M and D are found again in S. 5x" Stevin, who expressed our -- - z 2 in this manner.' 4 y 5D sec M ter , where sec and ter signify the "second" and "third" unknown quantity. The inverted letter Q is used to indicate division by Gallimard, 6 as in "12 Q 4 = 3" and "a 2 6 2 <I a 2 ." In 1790 Da Cunha 6 uses the horizontal letter TJ as a mark for division. 237. Rahris notation. In 1659 the Swiss Johann Heinrich Rahn published an algebra 7 in which he introduced -r- as a sign of division ( 194). Many writers before him had used ^ as a minus sign ( 164, 208). Rahn's book was translated into English by Thomas Brancker (a graduate of Exeter College, Oxford) and published, with additions from the pen of Joh. Pell, at London in 1668. Rahn's Teutsche Algebra was praised by Leibniz 8 as an "elegant algebra," nevertheless it did not enjoy popularity in Switzerland and the symbol -f- for division 1 Michael Stifel, Arithmetica Integra (Nurnberg, 1544), fol. 317F, 318r. This reference is taken from J. Tropfke, op. tit., Vol. II (2d ed., 1921 ), p. 28, n. 114. 2 Joseph Moxon, Mathematical Dictionary (3ded.; London, 1701), p. 190, 191. 3 Michael Stifel, Deutsche Arithmetica (Nurnberg, 1545), fol. 74^. We draw this information from J. Tropfke, op. cit., Vol. II (2d ed., 1921), p. 21. 4 S. Stevin, (Euvres (ed. A. Girard, 1634), Vol. I, p. 7, def . 28. 6 J. E. Gallimard, La Science du calcul numerique, Vol. I (Paris, 1751), p. 4; Methvde ... d'arithmetique, d'algebre et de geometric (Paris, 1753), p. 32. 6 J. A. da Cunha, Principles mathematicos (1790), p. 214. 7 J. H. Rahn, Teutsche Algebra (Zurich, 1659). 8 Leibnizens mathematische Schriften (ed. C. I. Gerhardt), Vol. VII, p. 214. DIVISION AND RATIO 271 was not adopted by his countrymen. In England, the course of events was different. The translation met with a favorable reception; Rahn's -f- and some other symbols were adopted by later English writers, and came to be attributed, not to Rahn, but to John Pell. It so happened that Rahn had met Pell in Switzerland, and had received from him (as Rahn informs us) the device in the solution of equations of dividing the page into three columns and registering the successive steps in the solution. Pell and Brancker never claimed for themselves the introduction of the -f- and the other symbols occurring in Rahn's book of 1569. But John Collins got the impression that not only the three-column arrangement of the page, but all the new algebraic symbols were due to Pell. In his extensive correspondence with John Wallis, Isaac Barrow, and others, Collins repeatedly spoke of -f- as "Pell's symbol." There is no evidence to support this claim ( 194) . 1 . The sign -r- as a symbol for division was adopted by John Wallis and other English writers. It came to be adopted regularly in Great Britain and the United States, but not on the European Continent. The only text not in the English language, known to us as using it, is one published in Buenos Aires; 2 where it is given also in the modified form /, as in f / 8=f&. In an American arithmetic, 3 the abbreviation -i-rs was introduced for "divisors," and -s-nds for "dividends," but this suggestion met with no favor on the part of other writers. 238. Leibniz' notations. In the Dissertatio de arte combinatoria (1668) 4 G. W. Leibniz proposed for division the letter C, placed hori- zontally, thus O, but he himself abandoned this notation and in- troduced the colon. His article of 1684 in the Ada eruditorum contains for the first time in print the colon ( : ) as the symbol for di- vision. 5 Leibniz says: ". . . . notetur, me divisionem hie designare hoc /r modo: x:y, quod idem est ac x divis. per y seu - ." In a publication of a the year 1710 8 we read: "According to common practice, the division 1 F. Cajori, "Rahn's Algebraic Symbols," Amer. Math. Monthly, Vol. XXXI (1924), p. 65-71. 2 Florentine Garcia, El aritme'tico Argentina (5th ed.; Buenos Aires, 1871), p. 102. The symbol ~ and its modified form are found in the first edition of this book, which appeared in 1833. 8 The Columbian Arithmetician, "by an American' 1 (Haverhill [Mass.], 1811), p. 41. 4 Leibniz, Opera omnia, Tom. II (Geneva, 1768), p. 347. 6 See Leibnizew mathematische Schriften (ed. C. I. Gerhardt), Vol. V (1858), p. 223. See also M. Cantor, Gesch. d. Mathematik, Vol. Ill (2d ed.; Leipzig), p. 194. 6 Miscellanea Berolinensia (Berlin, 1710), p. 156. See our 198. 272 A HISTORY OF MATHEMATICAL NOTATIONS is sometimes indicated by writing the divisor beneath the dividend, with a line between them; thus a divided by 6 is commonly indicated by r/'Very often however it is desirable to avoid this and to continue on the same line, but with the interposition of two points; so that a:b means a divided by b. But if, in the next place a : b is to be divided by c, one may write a : 6, : c, or (a : b) : c. Frankly, however, in this case the relation can be easily expressed in a different manner, namely a: (be) or a:bc, for the division cannot always be actually carried out but often can only be indicated and then it becomes necessary to mark the course of the deferred operation by commas or parentheses." In Germany, Christian Wolf was influential through his textbooks in spreading the use of the colon (:) for division and the dot () for multiplication. His influence extended outside Germany. A French translation of his text 1 uses the colon for division, as in "(ab):b." He writes: "a:mac = b:mbc." 239. In Continental Europe the Leibnizian colon has been used for division and also for ratio. This symbolism has been adopted in the Latin countries with only few exceptions. In 1878 Balbontin 2 used in place of it the sign -f- preferred by the English-speaking countries. Another Latin-American writer 3 used a slanting line in this manner, (y\3 1 =-^-- = = and also 12\3 = 4. An author in Peru 4 indicates division by writing the dividend and divisor on the same line, but inclosing the former in a parenthesis. Accordingly, "(20)5" meant 20-^5. Sometimes he uses brackets and writes the proportion 2:11 = 20:15 in this manner: "2:1[1J2: :20:15." 240. There are perhaps no symbols which are as completely ob- servant of political boundaries as are -f- and : as symbols for division. The former belongs to Great Britain, the British dominions, and the United States. The latter belongs to Continental Europe and the Latin-American countries. There are occasional authors whose prac- 1 C. Wolf, Cours de mathematique, Tom. I (Paris, 1747), p. 110, 118. 2 Juan Maria Balbontin, Tratado elemental de arilmetica (Mexico, 1878), p. 13. 3 Felipe Senillosa, Tratado elemental de ariftmetica (neuva ed.; Buenos Aires, 1844), p. 16. We quote from p. 47: "Este signo deque hemos hecho uso en la partition (\) no es usado generalmente; siendo el que se usa los dos punctos (:) 6 la forma de quebrado. Pero un quebrado denota mas bien un cociente 6 particion ejecutada que la operacion 6 acto del partir; asf hemos empleado este signo \ con analogia al del multiplicar que es e*ste: X." 4 Juan de Dios Salazar, Lecciones de aritmetica (Arequipa, 1827), p. v, 74, 89. DIVISION AND RATIO 273 tices present exceptions to this general statement of boundaries, but their number is surprisingly small. Such statements would not apply to the symbolisms for the differential and integral calculus, not even for the eighteenth century. Such statements would not apply to trigonometric notations, or to the use of parentheses or to the desig- nation of ratio and proportion, or to the signs used in geometry. Many mathematical symbols approach somewhat to the position of world-symbols, and approximate to the rank of a mathematical world-language. To this general tendency the two signs of division -r- and : mark a striking exception. The only appearance of -f- signi- fying division that we have seen on the European Continent is in an occasional translation of an English text, such as Colin Maclaurin's Treatise of Algebra which was brought out in French at Paris in 1753. Similarly, the only appearance of : as a sign for division that we have seen in Great Britain is in a book of 1852 by T. P. Kirkman. 1 Saverien 2 argues against the use of more than one symbol to mark a given operation. "What is more useless and better calculated to disgust a beginner and embarrass even a geometer than the three expressions -, :, -f-, to mark division?" 241. Relative position of divisor and dividend. In performing the operation of division, the divisor and quotient have been assigned various positions relative to the dividend. When the "scratch method" of division was practiced, the divisor was placed beneath the dividend and moved one step to the right every time a new figure of the quotient was to be obtained. In such cases the quotient was usually placed immediately to the right of the dividend, but some- times, in early writers, it was placed above the dividend. In short division, the divisor was often placed to the left of the dividend, so that a)b(c came to signify division. A curious practice was followed in the Dutch journal, the Maan- delykse Mathematische Liefhebberye (Vol. I [1759], p. 7), where a) signifies division by a, and (a means multiplication by a. Thus: ** />-)/ 7) /7_1_7 " M \ xy u a-f.c _ba+x * ~~ x James Thomson called attention to the French practice of writing the divisor on the right. He remarks: "The French place the divisor 1 T. P. Kirkman, First Mnemonial Lessons in Geometry, Algebra and Trigo- nometry (London, 1852). 2 Alexandre Saverien, Dictionnaire universel de maihematique et de physique (Paris, 1753), "Caractere." 274 A HISTORY OF MATHEMATICAL NOTATIONS to the right of the dividend, and the quotient below it This mode gives the work a more compact and neat appearance, and pos- sesses the advantage of having the figures of the quotient near the divisor, by which means the practical difficulty of multiplying the divisor by a figure placed at a distance from it is removed This method might, with much propriety, be adopted in preference to that which is employed in this country." 1 The arrangement just described is given in Bzout's arithmetic, 2 in the division of 14464 by 8, as follows: "144641 8_ |i808 ' 242. Order of operations in terms containing both -f- and X . If an arithmetical or algebraical term contains -f- and X , there is at present no agreement as to which sign shall be used first. "It is best to avoid such expressions. " 3 For instance, if in 24-7-4X2 the signs are used as they occur in the order from left to right, the answer is 12; if the sign X is used first, the answer is 3. Some authors follow the rule that the multiplications and divi- sions shall be taken in the order in which they occur. 4 Other textbook writers direct that multiplications in any order be performed first, then divisions as they occur from left to right. 5 The term a~bXb is interpreted by Fisher and Schwatt 6 as (a-r6)X&. An English com- mittee 7 recommends the use of brackets to avoid ambiguity in such cases. 243. Critical estimates of : and -r- as symbols. D. Andre 8 expresses himself as follows: "The sign : is a survival of old mathematical no- tations; it is short and neat, but it has the fault of being symmetrical toward the right and toward the left, that is, of being a symmetrical sign of an operation that is asymmetrical. It is used less and less. 1 James Thomson, Treatise on Arithmetic (18th ed.; Belfast, 1837). 2 Arithmetique de Bezout ... par F. Peyrard (13th ed.; Paris, 1833). 3 M. A. Bailey, American Menial Arithmetic (New York, 1892), p. 41. 4 Hawkes, Luby, and Teuton, First Course of Algebra (New York, 1910), p. 10. 5 Slaught and Lennes, High School Algebra, Elementary Course (Boston, 1907), p. 212. 6 G. E. Fisher and I. J. Schwatt, Text-Book of Algebra (Philadelphia, 1898), p. 85. 7 "The Report of the Committee on the Teaching of Arithmetic in Public Schools," Mathematical Gazette, Vol. VIII (1917), p. 238. See also p. 290. 8 Desire* Andre*, Des Notations mathtmatiques (Paris, 1909), p. 58, 59. DIVISION AND RATIO 275 .... When it is required to write the quotient of a divided by b, in the body of a statement in ordinary language, the expression a: b really offers the typographical advantage of not requiring, as does v-, a wider separation of the line in which the sign occurs from the two lines which comprehend it." In 1923 the National Committee on Mathematical Requirements 1 voiced the following opinion: "Since neither -7- nor :, as signs of di- vision, plays any part in business life, it seems proper to consider only the needs of algebra, and to make more use of the fractional form and (where the meaning is clear) of the symbol /, and to drop the symbol -7- in writing algebraic expressions. " 244. Notations for geometrical ratio. William Oughtred intro- duced in his Clavis mathematicae the dot as the symbol for ratio ( 181). He wrote ( 186) geometrical proportion thus, a.b::c.d. This nota- tion for ratio arid proportion was widely adopted not only in England, but also on the European Continent. Nevertheless, a new sign, the colon (:), made its appearance in England in 1651, only twenty years after the first publication of Oughtred 's text. This colon is due to the astronomer Vincent Wing. In 1649 he published in London his Urania practica, which, however, exhibits no special symbolism for ratio. But his Harmonicon cocleste (London, 1651) contains many times Oughtred's notation A.B::C .D, and many times also the new notation A : B : :C:D, the two notations being used interchangeably. Later there appeared from his pen, in London, three books in one volume, Logistica astronomica (1656), Doctrina spherica (1655), and Doctrina theorica (1655), each of which uses the notation A:B: : C:D. A second author who used the colon nearly as early as Wing was a schoolmaster who hid himself behind the initials "R.B." In his book entitled An Idea of Arithmetik, at first designed for the use of "the Free Schoole at Thurlow in Suffolk .... by R.B., Schoolmaster there" (London, 1655), one finds 1.6: :4.24 and also A:a: :C:c. W. W. Beman pointed out in Ulntermidiaire des mathematiciens, Volume IX (1902), page 229, that Oughtred's Latin edition of his Trigonometria (1657) contains in the explanation of the use of the tables, near the end, the use of : for ratio. It is highly improbable that the colon occurring in those tables was inserted by Oughtred himself. In the Trigonometria proper, the colon does not occur, and Ought- 1 Report of the National Committee on Mathematical Requirements under the Auspices of the Mathematical Association of America, Inc. (1923), p. 81. 276 A HISTORY OF MATHEMATICAL NOTATIONS red's regular notation for ratio and proportion A .B: :C .D is followed throughout. Moreover, in the English edition of Oughtred's trigo- nometry, printed in the same year (1657), but subsequent to the Latin edition, the passage of the Latin edition containing the : is recast, the new notation for ratio is abandoned, and Oughtred's notation is introduced. The : used to designate ratio ( 181) in Oughtred's Opuscula mathematica hactenus inedita (1677) may have been intro- duced by the editor of the book. It is worthy of note, also, that in a text entitled Johnsons Arith- metik; In two Bookes (2d ed.; London, 1633), the colon (:) is used to designate a fraction. Thus f is written 3:4. If a fraction be con- sidered as an indicated division, then we have here the use of : for division at a period fifty-one years before Leibniz first employed it for that purpose in print. However, dissociated from the idea of a frac- tion, division is not designated by any symbol in Johnson's text. In dividing 8976 by 15 he writes the quotient "598 6: 15." As shown more fully elsewhere ( 258), the colon won its way as the regular symbol for geometrical ratio, both in England and the European Continent. 245. Oughtred's dot and Wing's colon did not prevent experi- mentation with other characters for geometric ratio, at a later date. But none of the new characters proposed became serious rivals of the colon. Richard Balam, 1 in 1653, used the colon as a decimal separatrix, and proceeded to express ratio by turning the colon around so that the two dots became horizontal; thus "3 . . 1" meant the geometrical ratio 1 to 3. This designation was used by John Kirkby 2 in 1735 for arithmetical ratio; he wrote arithmetical proportion "9.. 6 = 6 . . 3." In the algebra of John Alexander, 3 of Bern, geometrical ratio is expressed by a dot, a. b, and also by a 6. Thomas York 4 in 1687 wrote a geometrical proportion "33600 7 : : 153600 32," using no sign at all between the terms of a ratio. In the minds of some writers, a geometrical ratio was something more than an indicated division. The operation of division was asso- ciated with rational numbers. But a ratio may involve incomrnensu- 1 Richard Bulam, Algebra: or The Doctrine of Composing, Inferring, and Re- solving an Equation (London, 1653), p. 4. 2 John Kirkby, Arithmetical Institutions (London, 1735), p. 28. 3 Synopsis algebraica, opus posthumum lohannis Alexandri, Bernatis-Helvetii. In usum scholae mathematical apud Hospitium-Christi Londinense (London, 1693), p. 16, 55. An English translation by Sam. Cobb appeared at London in 1709. 4 Thomas York, Practical Treatise of Arilhmetik (London, 1687), p. 146. DIVISION AND RATIO 277 rable magnitudes which are expressible by two numbers, one or both of which are irrational. Hence ratio and division could not be marked by the same symbol. Oughtred's ratio a.b was not regarded by him as an indicated division, nor was it a fraction. In 1696 this matter was taken up by Samuel Jeake 1 in the following manner: "And so by some, to distinguish them [ratios] from Fractions, instead of the in- tervening Line, two Pricks are set; and so the Ratio Sesquialtera 3 is thus expressed ." Jeake writes the geometrical proportion, z " 7 ( ) Emanuel Swedenborg starts out, in his Daedalus Hyperboreus (Upsala, 1716), to designate geometric proportion by : : : :, but on page 126 he introduces as a signum analogicum which is really used as a symbol for the ratio of quantities. On the European Continent one finds Herigone 2 using the letter TT to stand for "proportional" or ratio; he writes IT where we write : . On the other hand, there are isolated cases where : was assigned a different usage; the Italian L. Perini 3 employs it as separatrix between the number of feet and of inches; his "11:4" means 11 feet 4 inches. 246. Discriminating between ratio and division, F. Schmeisser 4 in 1817 suggested for geometric ratio the symbol . . , which (as previ- ously pointed out) had been used by Richard Balam, and which was employed by Thomas Dil worth 5 in London, and in 1799 by Zachariah Jess, 6 of Wilmington, Delaware. Schmeisser comments as follows: "At one time ratio was indicated by a point, as in a. 6, but as this signifies multiplication, Leibniz introduced two points, as in a:&, a designation indicating division and therefore equally inconvenient, and current only in Germany. For that reason have Monnich, v. Winterfeld, Krause and other thoughtful mathematicians in more recent time adopted the more appropriate designation a. .6." Schmeisser writes (p. 233) the geometric progression: "-J-3..6..12 ..24.. 48.. 96 ..... " 1 Samuel Jeake, AOriSTIKITAOrf A, or Arithmetick (London, 1696), p. 410. 2 Peter Herigone, Cursus mcdhematicus, Vol. I (Paris, 1834), p. 8. 3 Lodovico Perini, Gcometria pralica (Venezia, 1750), p. 109. 4 Fricdrich Schmeisser, Lehrbuch der reinen Mathesis, Erster Thcil, "Die Arithmetik" (Berlin, 1817), Vorrede, p. 58. 5 Thomas Dilworth, The Schoolmaster's Assistant (2d ed.; London, 1784). (First edition, about 1744.) 6 Zachariah Jess, System of Practical Surveying (Wilmington, 1799), p. 173. 278 A HISTORY OF MATHEMATICAL NOTATIONS Similarly, A. E. Layng, 1 of the Stafford Grammar School in j^ England, states: 'The Algebraic method of expressing a ratio -~ being a very convenient one, will also be found in the Examples, where it should be regarded as a symbol for the words the ratio of A to B, and not as implying the operation of division; it should not be used for book-work." 247. Division in the algebra of complex numbers. As, in the alge- bra of complex numbers, multiplication is in general not commu- tative, one has two cases in division, one requiring the solution of a = bx, the other the solution of a = yb. The solution of a = bx is designated by Peirce 2 r--, by Schroder 3 -,-, by Study 4 and Cartan =-. The solution of a = yb is designated by Peirce - ^ and by Schroder X o a: 6, by Study and Cartan -T. The X and the . indicate in this nota- tion the place of the unknown factor. Study and Cartan use also the notations of Peirce and Schroder. SIGNS OF PROPORTION 248. Arithmetical and geometrical progression. The notation -H- was used by W. Oughtred ( 181) to indicate that the numbers follow- ing were in continued geometrical proportion. Thus, -if- 2, 6, 18, 54, 162 are in continued geometric proportion. During the seventeenth and eighteenth centuries this symbol found extensive application; beginning with the nineteenth century the need of it gradually passed away, except among the Spanish-American writers. Among the many English writers using -ff are John Wallis, 5 Richard Sault 6 , Edward Cocker, 7 John Kersey, 8 William Whiston, 9 Alexander Mal- 1 A. E. Layng, Euclid's Elements of Geometry (London, 1891), p. 219. 2 B. Peirce, Linear Associative Algebra (1870), p. 17; Amer. Jour, of Math., Vol. IV (1881), p. 104. 8 E. Schroder, Formate Elemente der ahsoluten Algebra (Progr. Bade, 1874). 4 E. Study and E. Cartan, Encyclopedic des scien. math., Tom. I, Vol. I (1908), p. 373. * Phil. Trans., Vol. V (London, 1670), p. 2203. 6 Richard Sault, A New Treatise of Algebra (London [no date]). 7 Cocker's Artificial Arithmetick, by Edward Cocker, perused and published by John Hawkes (London, 1684), p. 278. 8 John Kersey, Elements of Algebra (London, 1674), Book IV, p. 177. 9 A. Tacquet's edition of W. Whiston 1 s Elemenla Euclidea geometriae (Amster- dam, 1725), p. 124. PROPORTION 279 colm, 1 Sir Jonas Moore, 2 and John Wilson. 3 Colin Maclaurin indi- cates in his Algebra (1748) a geometric progression thus: "-^-liqiq 2 : (f:q*:(f\ etc." E. Bezout 4 and L. Dcspiau 5 write for arithmetical progression "-v-1.3.5.7.9," and "4f3:6:12" for geometrical pro- gression. Symbols for arithmetic progression were less common than for geometric progression, and they were more varied. Oughtred had no symbol. Wallis 6 denotes an arithmetic progression A, B, C, D ~f, or by a, 6, c, d, e, /^f. The sign -f-, which we cited as occurring in Bezout and Despiau, is listed by Saverien 7 who writes " -7-1. 2. 3. 4. 5, etc." But Saverien gives also the six dots :::, which occur in Stone 8 and Wilson. 9 A still different designation, -, for arithmetical pro- gression is due to Kirkby 10 and Emerson, 11 another -Hr- to Clark, 12 again another -fr is found in Blassiere. 13 Among French writers using -~ for arithmetic progression and -H- for geometric progression are Lamy, 14 De Belidor, 15 Suzanne, 16 and Fournier; 17 among Spanish-American 1 Alexander Malcolm, A New System of Arithmetick (London, 1730), p. 115. 2 Sir Jonas Moore, Arithmetick in Four Books (3d ed.; London, 1688), begin- ning of the Book IV. 3 John Wilson, Trigonometry (Edinburgh, 1714), p. 24. 4 E. Bezout, Cours de mathtmatiques, Tome I (2. 6d.; Paris, 1797), "Arith- m6tiquc," p. 130, 165. 6 Select Amusements in Philosophy of Mathematics .... translated from the French of M. L. Dcspiau, Formerly Professor of Mathematics and Philosophy at Paris Recommended .... by Dr. Hutton (London, 1801), p. 19, 37, 43. 6 John Wallis, Operum mathematicorvm Pars Prima (Oxford, 1657), p. 230, 236. 7 A. Saverien, Dictionnaire universel de mathematique et de physique (Paris, 1753), art. "Caractere." 8 E. Stone, New Mathematical Dictionary (London, 1726), art. "Characters." 9 John Wilson, Trigonometry (Edinburgh, 1714). 10 John Kirkby, Arithmetical Institutions containing a compleat System of Arithmetic (London, 1735), p. 36. u W. Ernerson, Doctrine of Proportion (1763), p. 27. w Gilbert Clark, Oughtredus explicatus (London, 1682), p. 114. 13 J. J. Blassiere, Institution du calcul numerique et litteral (a La Haye, 1770), end of Part II. 14 Bernard Lamy, Siemens des mathematiques (3d ed. ; Amsterdam, 1692), p. 156. 18 B. F. de Belidor, Nouveau Cours de mathematique (Paris, 1725), p. 71, 139. 16 H. Suzanne, De la Maniere d' eludier 1 , es Mathematiques (2. &L; Paris, 1810), p. 208. C. F. Fournier, Elements d' Arithmttique et d'Algebre, Vol. II (Nantes, 1822). 280 A HISTORY OF MATHEMATICAL NOTATIONS writers using these two symbols are Senillosa, 1 Izquierdo, 2 Lidvano, 3 and Porfirio da Motta Pegado. 4 In German publications ~ for arith- metical progression and ^f for geometric progression occur less fre- quently than among the French. In the 1710 publication in the Mis- cellanea Berolinensia 5 -ff is mentioned in a discourse on symbols ( 198). The -=f was used in 1716 by Emanuel Swedenborg. 6 Emerson 7 designated harmonic progression by the symbol -^v and harmonic proportion by .V. . 249. Arithmetical proportion finds crude symbolic representation in the Arithmetic of Boethius as printed at Augsburg in 1488 (see Figure 103). Being, in importance, subordinate to geometrical pro- portion, the need of a symbolism was less apparent. But in the seven- teenth century definite notations came into vogue. William Oughtred appears to have designed a symbolism. Oughtred's language (Clavis [1652], p. 21) is "Ut 7.4:12.9 vel 7.7-3:12.12-3. Arithmetics proportionales sunt." As later in his work he does not use arithmetical proportion in symbolic analysis, it is not easy to decide whether the symbols just quoted were intended by Oughtred as part of his alge- braic symbolism or merely as punctuation marks in ordinary writing. John Newton 8 says: "As 8,5:6,3. Here 8 exceeds 5, as much as 6 exceeds 3." Wallis 9 says: "Et pariter 5,3; 11,9; 17,15; 19,17. sunt in cadcm progressione arithmetica." In P. Chelucci's 10 Inslitutiones analyticae, arithmetical proportion is indicated thus: 6.8'. '10. 12. Oughtred's notation is followed in the article "Caractere" of the Encydop6die Felipe Senillosa, Tratado elemental de Arismdica (Ncuva ed.; Buenos Aires, 1844), p. 46. 2 Gabriel Izquierdo, Tratado de Aritmetica (Santiago [Chile], 1859), p. 167. 3 Indalecio Lidvano, Tratado de Aritmetica (2. e\L; Bogota, 1872), p. 147. 4 Luiz Porfirio da Motta Pegado, Tratade elementar de arithmetica (2. e*d. ; Lisboa, 1875), p. 253. 6 Miscellanea Berolinensia (Berolini, 1710), p. 159. 6 Emanuel Swedberg, Daedalus hyperboreus (Upsala, 1716), p. 126. Facsimile reproduction in Kungliga Vetenskaps Societetens i Upsala Tvahundraarsminne (Upsala, 1910). 7 W. Emerson, Doctrine of Proportion (London, 1763), p. 2. 8 John Newton, Institutio mathematica or mathematical Institution (London, 1654), p. 125. 9 John Wallis, op. cit. (Oxford, 1657), p. 229. 10 Paolino Chelucci, Institutiones analyticae (editio post tertiam Romanam prima in Germania; Vienna, 1761), p. 3. See also the first edition (Rome, 1738), p. 1-15. PROPORTION 281 mtthodique (Mathtmatiques) (Paris: Lige, 1784). Lamy 1 says: "Proportion arithm&ique, 5,7 Y 10,12.c'est & dire qu'il y a mme difference entre 5 et 7, qu'entre 10 et 12." In Arnauld's geometry 2 the same symbols are used for arithmeti- cal progression as for geometrical progression, as in 7.3:: 13.9 and 6.2::12.4. Samuel Jeake (1696) 3 speaks of " Three Pricks or Points, some- times in disjunct proportion for the words is as." A notation for arithmetical proportion, noticed in two English seventeenth-century texts, consists of five dots, thus ::; Richard Balam 4 speaks of "arithmetical disjunct proportionals" and writes "2.4 :-:3.5"; Sir Jonas Moore 5 uses :: and speaks of "disjunct pro- portionals." Balam adds, "They may also be noted thus, 2... 4 = 3... 5." Similarly, John Kirkby 6 designated arithmetrical propor- tion in this manner, 9.. 6 = 6.. 3, the symbolism for arithmetical ratio being 8. .2. L'Abb6 Deidier (1739) 7 adopts 20. 2. \78.60. Be- fore that Weigcl 8 wrote "(o) 3| V 4.7" and "(o) 2.| V 3.5." Wolff (1710), 9 Panchaud, 10 Savcrien, 11 L'AbbS Foucher, 12 Emerson, 13 place 1 B. Lamy, Elemens dea mathemotiques (3. 6d.; Amsterdam, 1692), p. 155. 2 Antoine Arnauld, Nouveaux elemem de geometric (Paris, 1667); also in the edition issued at The Hague in 1690. Samuel Jeake, AOriSTIKIIAOriA or Arithmetick (London, 1696; Preface, 1674), p. 10-12. 4 Richard Balam, Algebra: or the Doctrine of Composing, Inferring, and Re- solving an Equation (London, 1653), p. 5. 5 Sir Jonas Moore, Moore's Arithmetick: In Four Books (3d ed.; London, 1688), the beginning of Book IV. 8 Rev. Mr. John Kirkby, Arithmetical Institutions containing a compleat Sys- tem of Arithmetic (London, 1735), p. 27, 28. 7 L'Abbc" Deidier, L'Arithm&iques des geometres t ou nouveau elemens de malhe- matiques (Paris, 1739), p. 219. 8 Erhardi Weigelii Specimina novarum inveniionum (Jenae, 1693), p. 9. 9 Chr. v. Wolff, Anfangsgrunde oiler math. Wissenschaften (1710), Vol. I, p. 65. See J. Tropfke, op. cit., Vol. Ill (2d ed., 1922), p. 12. 10 Benjamin Panchaud, Entreliens ou lecons mathematiques, Premier Parti (Lausanne et Geneve, 1743), p. vii. 11 A. Saverien, Dictionnaire universel (Paris, 1753), art. "Proportion arith- metique." 11 L'Abb6 Foucher, Geometrie metaphysique ou essai d f analyse (Paris, 1758), p. 257 W. Emerson, The Doctrine of Proportion (London, 1763), p. 27. 282 A HISTORY OF MATHEMATICAL NOTATIONS the three dots as did Chelucci and Deidier, viz., a.b'.'c.d. Cosalli 1 writes the arithmetical proportion a : b V c:d. Later Wolff 2 wrote a 6 = c d. Blassiere 3 prefers 2:7-rrlO:15. Juan Gerard 4 transfers Oughtred's signs for geometrical proportion to arithmetical proportion and writes accordingly, 9.7:: 5.3. In French, Spanish, and Latin- Ameri- can texts Oughtred's notation, 8.6:5.3, for arithmetical proportion has persisted. Thus one finds it in Benito Bails, 6 in a French text for the military, 6 in Fournier, 7 in Gabriel Izquierdo, 8 in Indalecio Lievano. 9 250. Geometrical proportion. A presentation of geometrical pro- portion that is not essentially rhetorical is found in the Hindu Bakh- shilli arithmetic, where the proportion 10: J C V = 4: }f is written in the form 10 10 1 163 60 4 1 pha 163 150 It was shown previously ( 124) that the Arab al-Qalasddi (fifteenth century) expresses the proportion 7.12 = 84:144 in this manner: 144 /. 84 /. 12 /. 7. Regiomontanus in a letter writes our modern a : b : c in the form a.b.c, the dots being simply signs of separation. 11 In the edition of the Arithmetica of Boethius, published at Augsburg in 1488, a crude representation of geometrical and arithmetical propor- 1 Scritli inedili del P. D. Pielro Cossali .... pubblicati da B. Boncompagni (Rome, 1857), p. 75. 2 Chr. v. Wolff., op. cit. (1750), Vol. I, p. 73. 3 J. J. Blassiere, Institution du calcul numerique et lateral (a La Haye 1770), the end of Part II. 4 Juan Gerard, Tratado complete de aritmetica (Madrid, 1798), p. 69. 6 Benito Bails, Principios de matematica de la real academia de San Fernando (2. ed.), Vol. I (Madrid, 1788), p. 135. 6 Cours de mathSmaliques, d V usage des ecoles imperiales militaires ... r6dige" par ordre de M. le Ge'ne'ral de Division Bellavene ... (Paris, 1809), p. 52. Dedica- tion signed by "Allaize, Billy, Puissant, Boudrot, Professeurs de mathdmatiques a TEcole de Saint-Cyr." 7 C. F. Fournier, Elements d' arithmetique et d'algebre. Tome II (Nantes, 1842), p. 87. 8 Gabriel Izquierdo, op. cit. (Santiago [Chile], 1859), p. 155. 9 Indalecio Li6vano, Tratado aritmetica (2d ed.; Bogota, 1872), p. 147. 10 G. R. Kaye, The Bakhshall Manuscript, Parts I and II (Calcutta, 1927), p. 119. " M. Curtze, Abhandlungen z. Geschichte d. Mathematik, Vol. XII (1902), p. 253. PROPORTION 283 tion is given, as shown in Figure 103. The upper proportion on the left is geometrical, the lower one on the left is arithmetical. In the latter, the figure 8 plays no part; the 6, 9, and 12 are in arithmetical proportion. The two exhibitions on the right relate to harmonical and musical proportion. Proportion as found in the earliest printed arithmetic (in Treviso, <ttpttfciitf fen iraoitfct. o<wi nu mtoiu nunp &.a*fcermir w c. ttfoei.ii.faciA.'i-Qti riuAiperf.Oiiatuot Stf^idUcrfj^omone*. MiarfiM a fruofccnano v-^ > V^^X. SSgFS f f\ \ iil><fimnipUee. 44- H < I > I i I rtreirtrwc*inttiplKft: i V / F"^ aei&.r.qtionumro I S^ J rtrtCf mftlOtlf maiMifiB ftniitqr. PWT per rbwbtirtboUvi i _4 icrturimicxtmiaitnirtnatmkaim/ ougtiflcerceUct nominatUTunu*. . dttwficM confonannw. If Ktfqwirerti i,ppouion re^ui . t (I ntaiHa:ertcron ofoiiantul. Sejtfo ^.9.wl.8.ot>. a cM>arari rtt>t>ut a b c d c f ftinpbonli.|>uot)eemf o^ ftnav *44 Jimctf enrenu3K t par tn^iamulnplicate. f o t.f.irfi ?ir<i ft nxNj conRtterart jpocbou... - - - - u mugii.fl in mufi<o motm tooottcxaf.qii " coriifortojurmnfurdc . tooot vocof .qiit 0(0} mull/ d Jmoni* eft. ttftttftff. Untxnort cftq6 MaWeront bi Mnic sfononriarii io.to* biffrrfti %:lkui inter fcfcuiicrciit fcfqiMl/ nf foU < epocboiw Mf FIG. 103. From the Arithmetica of Boethius, as printed in 1488, the last two pages. (Taken from D. K. Smith's Rara arithmetica [Boston, 1898], p. 28.) 1487) is shown in Figure 39. Stifel, in his edition of Rudolff s Coss (1553), uses vertical lines of separation, as in "100 | i- z I 100 z | Facit J zz ." Tartaglia 1 indicates a proportion thus: <( Se X 3// val 4 // che valeranno 28." Chr. Clavius 2 writes: "9 . 126 . 5 . ? fiunt 70 ." 1 N. Tartaglia, La prima parte del General Tratato di Nvmeri, etc. (Venice 1556), fol. 1291?. 2 Chr. Clavius, Epitome arithmeticae practicae (Rome, 1583), p. 137. 284 A HISTORY OF MATHEMATICAL NOTATIONS This notation is found as late as 1699 in Corachan's arithmetic 1 in such statements as "A . B . C . D . 5 . 7 . 15 . 21 . " Schwenter 2 marks the geometric proportion 68 51 85, then finds the product of the means 51X85 = 4335 and divides this by 68. In a work of Galileo, 3 in 1635, one finds: "Regula aurea 58 95996 . 21600 . 21600 57597600 95996 191992 58 357 20735 3339 42" 13600 In other places in Galileo's book the three terms in the proportion are not separated by horizontal lines, but by dots or simply by spac- ing. Johan Stampioen, 4 in 1639, indicates our a:6 = 6:c by the sym- bolism: "a,, b gel : b c . Further illustrations are given in 221. These examples show that some mode of presenting to the eye the numbers involved in a geometric proportion, or in the applica- tion of the rule of three, had made itself felt soon after books on mathe- matics came to be manufactured. Sometimes the exposition was rhe- torical, short words being available for the writing of proportion. As late as 1601 Philip Lansberg 5 wrote "ut 5 ad 10; ita 10 ad 20," meaning 1 Ivan Bavtista Corachan, Arilhmetica demonslrada (Valencia, 1699), p. 199. 2 Daniel Schwentcr, Geomclriae practicae novae et auctac tractatus (Numbers, 1623), p. 89. 3 Syslema Cosmicvm, aucthore Galilaeo Galilaei Ex Italica lingua f aline conversum (Florence, 1635), p. 294. 4 Johan Stampioen, Algebra ofte nieuwe Slel-Regel (The Hague, 1639), p. 343. 6 Philip Lansberg, Triangulorum geomelriae libri quaiuor (Middelburg [Zee- land], 1663), p. 5. PROPORTION 285 5:10 = 10:20. Even later the Italian Cardinal Michelangelo Ricci 1 wrote "esto AC ad CB, ut 9 ad 6." If the fourth term was not given, but was to be computed from the first three, the place for the fourth term was frequently left vacant, or it was designated by a question mark. 251. Oughtred' s notation. As the symbolism of algebra was being developed and the science came to be used more extensively, the need for more precise symbolism became apparent. It has been shown ( 181) that the earliest noted symbolism was introduced by Ought- red. In his Clavis mathematicae (London, 1631) he introduced the notation 5 . 10 : : 6 . 12 which he retained in the later editions of this text, as well as in his Circles of Proportion (1632, 1633, 1660), and in his Trigonometria (1657). As previously stated ( 169) the suggestion for this symbolism may have come to Oughtred from the reading of John Dee's Introduction to Billingley's Euclid (1570). Probably no mathematical symbol has been in such great demand in mathematics as the dot. It could be used, conveniently, in a dozen or more different meanings. But the avoid- ance of confusion necessitates the restriction of its use. Where then shall it be used, and where must other symbols be chosen? Oughtred used the dot to designate ratio. That made it impossible for him to fol- low John Napier in using the dot as the scparatrix in decimal fractions. Oughtred could not employ two dots (:) for ratio, because the two dots were already pre-empted by him for the designation of aggre- gation, :A-\-B: signifying (A-{-B). Oughtred reserved the dot for the writing of ratio, and used four dots to separate the two equal ratios. The four dots were an unfortunate selection. The sign of equality ( = ) would have been far superior. But Oughtred adhered to his notation. Editions of his books containing it appeared repeatedly in the seventeenth century. Few symbols have met with more prompt adoption than those of Oughtred for proportion. Evidently the time was ripe for the introduction of a definite unambigu- ous symbolism. To be sure the adoption was not immediate. Nine- teen years elapsed before another author used the notation A.B:: C .D. In 1650 John Kersey brought out in London an edition of Edmund Wingate's Arithmetique made easie, in which this notation is used. After this date, the publications employing it became frequent, some of them being the productions of pupils of Oughtred. We have 1 Michaelis Angeli Rictii exercitatio geometrica de maximis et minimis (London, 1068), p. 3. 286 A HISTORY OF MATHEMATICAL NOTATIONS seen it in Vincent Wing, 1 Seth Ward, 2 John Wallis, 3 in "R.B.," a schoolmaster in Suffolk, 4 Samuel Foster, 5 Sir Jonas Moore, 6 and Isaac Barrow. 7 John Wallis 8 sometimes uses a peculiar combination of processes, involving the simplification of terms, during the very act of writing proportion, as in ' '%A = 4A . %A = 3 A : : %A = 2 A f A : : 8 . 6 : : 4.3." Here the dot signifies ratio. The use of the dot, as introduced by Oughtred, did not become universal even in England. As early as 1651 the astronomer, Vincent Wing ( 244), in his Harmonicon Coeleste (London), introduced the colon ( : ) as the symbol for ratio. This book uses, in fact, both nota- tions for ratio. Many times one finds A.B'.'.C.D and many times A:B::C:D. It may be that the typesetter used whichever notation happened at the moment to strike his fancy. Later, Wing published three books ( 244) in which the colon (:) is used regularly in writing ratios. In 1655 another writer, "R.B.," whom we have cited as using the symbols A.B::C.D, employed in the same publication also A:B::C:D. The colon was adopted in 1661 by Thomas Streete. 9 That Oughtred himself at any time voluntarily used the colon as the sign for ratio does not appear. In the editions of his Clavis of 1648 and 1694, the use of : to signify ratio has been found to occur only once in each copy ( 186) ; hence one is inclined to look upon this notation in these copies as printer's errors. 252. Struggle in England between Oughtred's and Wing's notations, before 1 700. During the second half of the seventeenth century there was in England competition between (.) and (:) as the symbols for the designation of the ratio ( 181, 251). At that time the dot main- tained its ascendancy. Not only was it used by the two most influ- 1 Vincent Wing, Harmonicon coeleste (London, 1651), p. 5. 2 Seth Ward, In Ismaelis Bullialdi astronomiae philolaicae fundamenla in- quisitio brevis (Oxford, 1653), p. 7. 3 John Wallis, Elenchus geometriae Hobbianac (Oxford, 1655), p. 48; Operum ma- thematicorum pars alter a (Oxford, 1656), the part on Arithmetica infinitorum, p. 181. 4 An Idea of ArUhmetick, at first designed for the use of the Free Schoole at Thurlow in Suffolk By R. B., Schoolmaster there (London, 1655), p. 6. 6 Miscellanies: or mathematical Lucrubations of Mr. Samuel Foster .... by John Twyden (London, 1659), p. 1. 6 Jonas Moore, Arilhmelick in two Books (London, 1660), p. 89; Moore's Arithmetique in Four Books (3d ed.; London, 1688), Book IV, p. 401. 7 Isaac Barrow's edition of Euclid's Data (Cambridge, 1657), p. 2. 8 John Wallis, Adversus Marci Meibomii de Proportionibus Dialogum (Oxford, 1657), "Dialogum," p. 54. 9 Thomas Streete, Astronomia Carolina (1661). See J. Tropfke, Geschichte der Elcmentar-Mathematik, 3. Bd., 2. Aufl. (Berlin und Leipzig, 1922), p. 12. PROPORTION 287 ential English mathematicians before Newton, namely, John Wallis and Isaac Barrow, but also by David Gregory, 1 John Craig, 2 N. Mercator, 3 and Thomas Brancker. 4 I. Newton, in his letter to Olden- burg of October 24, 1676, 5 used the notation . :: . , but in Newton's De analyxi per aequationes tcrminorum infinitas, the colon is employed to designate ratio, also in his Quadratures curvarum. Among seventeenth-century English writers using the colon to mark ratio are James Gregory, 6 John Collins, 7 Christopher Wren, 8 William Leybourn, 9 William Sanders, 10 John Hawkins, 11 Joseph Raphson, 12 E. Wells, 13 and John Ward. 14 253. Struggle in England between Oughtred's and Wing's notations during 1700-1750. In the early part of the eighteenth century, the dot still held its place in many English books, but the colon gained in ascendancy, and in the latter part of the century won out. The single dot was used in John Alexander's Algebra (in which proportion is written in the form a.biic.X and also in the form a~b:c~X) u and, in John Colson's translation of Agnesi (before 1760). 16 It was used 1 David Gregory in Phil. Trans., Vol. XIX (1G95-97), p. 645. 2 John Craig, Methodus fiyurarum lineis rectis et curvis (London, 1G85). Also his Traclatus mathematicus (London, 1G93), but in 1718 he often used : : : .in his De Calculo Fluentium Libri Duo, brought out in London. 3 N. Mercator, Logarithmotechnia (London, 1668), p. 29. 4 Th. Brancker, Introduction to Algebra (trans, of Rhonius; London, 1668), p. 37. 5 John Collins, Commercium epistolicum (London, 1712), p. 182. 6 James Gregory, Vera circuli et hyperbolae quadratura (Patavia, 1668), p. 33. 7 J. Collins, Mariners Plain Scale New Plain' d (London, 1659). 8 Phil. Trans., Vol. Ill (London), p. 868. 9 W. Leybourn, The Line of Proportion (London, 1673), p. 14. 10 William Sanders, Elementa geometriae (Glasgow, 1686), p. 3. 11 Cocker's Decimal Arithmetick .... perused by John Hawkins (London, 1695) (Preface dated 1684), p. 41. 12 J. Haphson, Analysis aequationum universalis (London, 1697), p. 26. 13 E. Wells, Elementa arithmeticae numerosae et speciosae (Oxford, 1698), p. 107. 14 John Ward, A Compendium of Algebra (2d ed.; London, 1698), p. 62. 15 A Synopsis of Algebra. Being the Posthumous Work of John Alexander, of Bern in Swisserland. To which is added an Appendix by Humfrey Ditton Done from the Latin by Sam. Cobb, M.A. (London, 1709), p. 16. The Latin edi- tion appeared at London in 1693. 16 Maria Gaetana Agnesi, Analytical Institutions, translated into English by the late Rev. John Colson Now first printed .... under the inspection of Rev. John Hellins (London, 1801). 288 A HISTORY OF MATHEMATICAL NOTATIONS by John Wilson 1 and by the editors of Newton's Universal arithmetick? In John Harris' Lexicon technicum (1704) the dot is used in some articles, the colon in others, but in Harris' translation 3 of G. Pardies* geometry the dot only is used. George Shelley 4 and Hatton 5 used the dot. 254. Sporadic notations. Before the English notations . : : . and : :: : were introduced on the European Continent, a symbolism con- sisting of vertical lines, a modification of Tartaglia's mode of writing, was used by a few continental writers. It never attained popularity, yet maintained itself for about a century. Ren6 Descartes (1619-21) appears to have been the first to introduce such a notation a|fe||c|d. In a letter 7 of 1638 he replaces the middle double stroke by a single one. Slusius 8 uses single vertical lines in designating four numbers in geometrical proportion, p \ a \ e \ d a. With Slusius, two vertical strokes 1 1 signify equality. Jaques de Billy 9 marks five quantities in continued proportion, thus 3 #5|#5 1| 2|#5+1 |3+#5, where & means "square root." In reviewing publications of Huygens and others, the original notation of Descartes is used in the Journal des Sgavans (Amsterdam) 10 for the years 1701, 1713, 1716. Likewise, Picard, 11 De la Hire, 12 Abraham de Graaf , 13 and Parent 14 use the notation a \ b \ \ xx \ ab. I John Wilson, Trigonometry (Edinburgh, 1714), p. 24. 2 1. Newton, Arithmetica universalis (ed. W. Whiston; Cambridge, 1707), p. 9; Universal Arithmetick, by Sir Isaac Newton, translated by Mr. llalphson .... revised .... by Mr. Cunn (London, 1769), p. 17. 3 Plain Elements of Geometry and Plain Trigonometry (London, 1701), p. 63. 4 G. Shelley, Wingate's Arithmetick (London, 1704), p. 343. 6 Edward Hatton, An Intire System of Arithmetic (London, 1721), p. 93. 6 (Euvres des Descartes (e*d. Adam et Tannery), Vol. X, p. 240. 7 Op. dt., Vol. II, p. 171. 8 Renati Francisci Slusii mesolabum seu duae mediae proportionates, etc. (1668), p. 74. See also Slusius' reply to Huygens in Philosophical Transactions (London), Vols. III-IV (1668-69), p. 6123. 9 Jaques de Billy, Nova geometriae clavis (Paris, 1643), p. 317. 10 Journal des S^avans (Amsterdam, ann6e 1701), p. 376; ibid. (annc*e 1713), p. 140, 387; ibid. (ann6e 1716), p. 537. II J. Picard in Memoires de I'Acadtmie r. des sciences (depuis 1666 jusqu'a 1699), Tome VI (Paris, 1730), p. 573. 12 De la Hire, Nouveaux elemens des sections coniques (Paris, 1701), p. 184. J. Tropfke refers to the edition of 1679, p. 184. 18 Abraham de Graaf, De vervulling van der geomelria en algebra (Amsterdam, 1708), p. 97. 14 A. Parent, Essais et recherches de mathemalique et de physique (Paris, 1713), p. 224. PROPORTION 289 It is mentioned in the article "Caractere" in Diderot's Encyclopedic (1754) . La Hire writes also "aa \ \ xx \ \ ab" for a 2 : re 2 = a: 2 : ab. On a subject of such universal application in commercial as well as scientific publications as that of ratio and proportion, one may expect to encounter occasional sporadic attempts to alter the symbolism. Thus Herigone 1 writes "hg TT ga 2|2 hb IT bd, signifi. HG est ad GA, vt HB ad BD," or, in modern notation, hg:ga hb:bd; here 2 1 2 signifies equality, TT signifies ratio. Again Peter Mengol, 2 of Bologna, writes "a;r:a2;ar" for a:r a?:ar. The London edition of the algebra of the Swiss J. Alexander 3 gives the signs . :: . but uses more often designa- tions like b~a: . Ade Mercastel, 4 of Rouen, writes 2,, 3 ;;8,,12. A close approach to the marginal symbolism of John Dee is that of the Spaniard Zaragoza 5 4.3:12.9. More profuse in the use of dots is TT J. Kresa 6 who writes x. . .r::r. . .-, also AE. .EF::AD. .DG. The latter form is adopted by the Spaniard Cassany 7 who writes 128. . 119 :: 3876; it is found in two American texts, 8 of 1797. In greater conformity with pre-Oughtredian notations is van Schooten's notation 9 of 1657 when he simply separates the three given numbers by two horizontal dashes and leaves the place for the unknown number blank. Using Stevin's designation for decimal Ib. flor. Ib. fractions, he writes "65 95,753 -1." Abraham de Graaf 10 is 1 Pierre Herigone, Cvrsvs mathematici (Paris, 1644), Vol. VI, "Explicatio notarum." The first edition appeared in 1642. 2 Pietro Mengoli, Geometriae speciosae elementa (Bologna, 1659), p. 8. 3 Synopsis algcbraica, Opus posthumum Johannis Alcxandri, Bcrnatis-Hel- vctii (London, 1693), p. 135. 4 Jean Baptiate Adrien de Mercastel, Arithmetique cUmontree (Rouen, 1733), p. 99. 6 Joseph Zaragoza, Arithmetica universal (Valencia, 1669), p. 48. 6 Jacob Kresa, Analysis speciosa trigonometriae sphericae (Prague, 1720), p. 120, 121. 7 Francisco Cassany, Arithmetica Deseada (Madrid, 1763), p. 102. 8 American Tutor's Assistant. By sundry teachers in and near Philadelphia (3d ed.; Philadelphia, 1797), p. 57, 58, 62, 91-186. In the "explanation of char- acters," : : : : is given. The second text is Chauncey Lee's American Accomptant (Lansingburgh, 1797), where one finds (p. 63) 3. .5: : 6. .10. 9 Francis a Schooten, Leydensis, Exercitationum mathematicarum liber primus (Leyden, 1657), p. 19. 10 Abraham de Graaf, De Geheele mathesia ofwiskonst (Amsterdam, 1694), p. 16. 290 A HISTORY OF MATHEMATICAL NOTATIONS partial to the form 2 4 = 612. Thomas York 1 uses three dashes 12542910?, but later in his book writes "33600 7 :: 153600 32," the ratio being here indicated by a blank space. To distinguish ratios from fractions, Samuel Jeake 2 states that by some authors "instead of the intervening Line, two Pricks are set; and so the Ratio 3 11 sesquialtera is thus expressed ." Accordingly, Jeake writes " ::9.7." In practical works on computation with logarithms, and in some arithmetics a rhetorical and vertical arrangement of the terms of a proportion is found. Mark Forster 3 writes: "As Sine of 40 deg. 9 , 8080675 To 1286 3,1092401 So is Radius 10,0000000 To the greatest Random 2000 3 , 301 1726 Or, For Random at 36 deg." As late as 1789 Benjamin Workman 4 writes " ^L^lnsj '" 255. Oughtred's notation on the European Continent. On the Euro- pean Continent the dot as a symbol of geometrical ratio, and the four dots of proportion, . :: ., were, of course, introduced later than in England. They were used by Dulaurens, 5 Prestet, 6 Varignon, 7 Pardies, 8 De THospital, 9 Jakob Bernoulli, 10 Johann Bernoulli, 11 Carr6, 12 Her- 1 Thomas York, Practical Treatise of Arithmetick (London, 1687), p. 132, 146. 2 Samuel Jeake, AOriSTIKHAOFlA, or Arithmetick (London, 1696 [Preface, 1674]), p. 411. 3 Mark Forster, Arithmetical Trigonometry (London, 1690), p. 212. 4 Benjamin Workman, American Accountant (Philadelphia, 1789), p. 62. 6 Francisci Dulaurens, Specimina mathematica (Paris, 1667), p. 1. 6 Jean Prestet, Siemens des mathematiques (Preface signed "J.P.") (Paris, 1675), p. 240. Also Nouveaux elemens des mathematiques, Vol. I (Paris, 1689), p. 355. 7 P. Varignon in Journal des S^avans, ann6e 1687 (Amsterdam, 1688), p. 644. Also Varignon, Eclair cissemens sur V analyse des infiniment petits (Paris, 1725), p. 16. 8 (Euvres du R. P. Ignace-Gaston Pardies (Lyon, 1695), p. 121. 9 De 1'Hospital, Analyse des infiniment petits (Paris, 1696), p. 11. 10 Jakob Bernoulli in Acta eruditorum (1687), p. 619 and many other places. 11 Johann Bernoulli in Histoire de V academic r. des sciences, ann6e 1732 (Paris, 1735), p. 237. 12 L. Carre", Methode pour la Mesure des Surfaces (Paris, 1700), p. 5. PROPORTION 291 maim, 1 and Rollc; 2 also by De Reaumur, 3 Saurin, 4 Parent, 5 Nicole, 6 Pitot, 7 Poleni, 8 De Mairan, 9 and Maupertuis. 10 By the middle of the eighteenth century, Oughtred's notation A.B::C .D had disappeared from the volumes of the Paris Academy of Sciences, but we still find it in textbooks of Belidor, 11 Guido Grandi, 12 Diderot, 13 Gallimard, 14 De la Chapelle, 15 Fortunato, 16 L'Abbe Foucher, 17 and of Abb Girault de Koudou. 18 This notation is rarely found in the writings of German authors. Erhard Weigel 19 used it in a philosophical work of 1693. Christian Wolf 20 used the notation "DC .AD:: EC .ME" in 1707, and in 1710 "3 . 12 : : 5 . 20" and also "3 : 12 = 5 : 20." Beguelin 21 used the dot for ratio in 1773. From our data it is evident that A.B::C .D began 1 J. Hermann in Ada eruditorum (1702), p. 502. 2 M. Rollc in Journal des S$avans, ann6e 1702 (Amsterdam, 1703), p. 399. 3 R. A. F. de Reaumur, Histoire de V academic r. des sciences, annc*e 1708 (Paris, 1730), "Me" moires," p. 209, but on p. 199 he used also the notation : : : :. 4 J. Saurin, op. ciL, ann6e 1708, "M6moires," p. 26. 5 Antoine Parent, op. cit., annee 1708, "Mcmoircs," p. 118. 6 F. Nicole, op. cit., amide 1715 (Paris, 1741), p. 50. 7 II. Pitot, op. tit., amide 1724 (Paris, 1726), "M6moires," p. 109. 8 Joannis Poleni, Epislolarvm mathcmaticarvm Fascicvlvs (Patavii, 1729). 9 J. J. de Mairan, Histoire de I'academie r. des sciences, anndc 1740 (Paris, 1742), p. 7. 10 P. L. Maupertuis, op. tit., ann6e 1731 (Paris, 1733), "M&noircs," p. 465. 11 B. F. de Belidor, Nouveau Cours de mathematique (Paris, 1725), p. 481. 12 Guido Grandi, Elementi geometrici piani e solide de Euclide (Florence, 1740). 13 Deny s Diderot, Memoir -es sur differens sujets de Mathematiques (Paris, 1748), p. 16. 14 J. E. Gallimard, Geometrie elementaire d 'Euclide (nouvelle 6d. ; Paris, 1749), p. 37. 16 De la Chapelle, Traite des sections coniqucs (Paris, 1750), p. 150. 16 F. Fortunato, Elementa matheseos (Brescia, 1750), p. 35. 17 L'Abb6 Foucher, Geometric metaphy$ique ou Essai d' analyse (Paris, 1758), p. 257. 18 L'Abb6 Girault de Koudou, Lemons analytiques du calcul des fluxions et des fluentes (Paris, 1767), p. 35. 19 Erhardi Weigelii Philosophia mathematica (Jenae, 1693), "Specimina no- varum inveiitionum," p. 6, 181. 20 C. Wolf in Ada eruditorum (1707), p. 313; Wolf, Anfangsgriinde aller mathe- matischen Wissemchaften (1710), Band I, p. 65, but later Wolf adopted the nota- tion of I^eibniz, viz., A:B C:D. See J. Tropfke, Geschichte der Elementar-Mathe- matik, Vol. Ill (2d ed.; Berlin und Leipzig, 1922), p. 13, 14. 21 Nicolas dc Beguelin in Nouvcaux mtmoires de I'academie r. des sciences et belles-lettres, annexe 1773 (Berlin, 1775), p. 211. 292 A HISTORY OF MATHEMATICAL NOTATIONS to be used in the Continent later than in England, and it was also later to disappear on the Continent. 256. An unusual departure in the notation for geometric propor- tion which involved an excellent idea was suggested by a Dutch author, Johan Stampioen, 1 as early as the year 1639. This was only eight years after Oughtrcd had proposed his . :: . Stampioen uses the designation A, ,# = C, ,D. We have noticed, nearly a century later, the use of two commas to represent ratio, in a French writer, Mercastel. But the striking feature with Stampioen is the use of Recorde's sign of equality in writing proportion. Stampioen antici- pates Leibniz over half a century in using = to express the equality of two ratios. He is also the earliest writer that we have seen on the European Continent to adopt Recorde's symbol in writing ordinary equations. He was the earliest writer after Descartes to use the ex- ponential form a 3 . But his use of = did not find early followers. He was an opponent of Descartes whose influence in Holland at that time was great. The employment of = in writing proportion appears again with James Gregory 2 in 1668, but he found no followers in this practice in Great Britain. 257. Slight modifications of Oughtred's notation. A slight modifica- tion of Oughtred's notation, in which commas took the place of the dots in designating geometrical ratios, thus A , B : : C , D, is occasionally encountered both in England and on the Continent. Thus Sturm 3 4bb 4b writes "36 , 26 : : 26, -^g- sive ~ /' Lamy 4 "3,6:: 4, 8," as did also Ozanam, 5 De Moivre, 6 David Gregory, 7 L'Abb6 Deidier, 8 Belidor, 9 who also uses the regular Oughtredian signs, Maria G. Agnesi, 10 1 Johan Stampioen d'Jonghe, Algebra ofte Nieuwe Stel-Regel ('& Graven-Have, 1639). 2 James Gregory, Geometriae Pars Vniversalis (Padua, 1668), p. 101. 3 Christopher Sturm in Ada erudilorum (Leipzig, 1685), p. 260. 4 R. P. Bernard Lamy, Elcmens dez mathematiques, troisieme edition revue ct augmented sur PimprismS a Paris (Amsterdam, 1692), p. 156. 5 J. Ozanam, Traite des lignes du premier genre (Paris, 1687), p. 8; Ozanam, Count de mathemalique, Tome III (Paris, 1693), p. 139. 6 A. dc Moivre in Philosophical Transactions, Vol. XIX (London, 1698), p. 52; De Moivre, Miscellanea analylica de seriebus (London, 1730), p. 235. 7 David Gregory, Ada eruditorum (1703), p. 456. 8 L'Abbe* Deidier, La Mesure des Surfaces el des Solides (Paris, 1740), p. 181. 9 B. F. de Belidor, op. tit. (Paris, 1725), p. 70. 10 Maria G. Agnesi, Inslituzioni analitiche, Tome I (Milano, 1748), p. 76. PROPORTION 293 Nicolaas Ypey, 1 and Manfredi. 2 This use of the comma for ratio, rather than the Oughtredian dot, does not seem to be due to any special cause, other than the general tendency observable also in the notation for decimal fractions, for writers to use the dot and comma more or less interchangeably. An odd designation occurs in an English edition of Ozanam, 3 namely, "A .2.5.3:: CA.D.6," where A,,C,Z> are quantities in geometrical proportion and the numbers are thrown in by way of concrete illustration. 258. The notation: :: : in Europe and America. The colon which replaced the dot as the symbol for ratio was slow in making its appear- ance on the Continent. It took this symbol about half a century to cross the British Channel. Introduced in England by Vincent Wing in 1651, its invasion of the Continent hardly began before the begin- ning of the eighteenth century. We find the notation A:B::C:D used by Leibniz, 4 Johann Bernoulli, 5 De la Hire, 6 Parent, 7 Bomie, 8 Saulmon, 9 Swedcnborg, 10 Lagny, 11 Senes, 12 Chevalier de Louville, 13 Clairaut, 14 Bouguer, 15 Nicole (1737, who in 1715 had used . :: .), 16 La 1 Nicolaas Ypey, Grondbeginselen der Keegelsneeden (Amsterdam, 1769), p. 3. 2 Gabriello Manfredi, De Constructione Aequationum differentialium primi (jradus (1707), p. 123. 3 J. Ozanam, Cursus mathematicus, translated "by several Hands" (London, 1712), Vol. I, p. 199. 4 Ada crudiiorum (1684), p. 472. 6 Johanne (I) Bernoulli in Journal des S^avans, ann6e 1698 (Amsterdam, 1709), p. 576. See this notation used also in Tann6e 1791 (Amsterdam, 1702), p. 371. 6 De la Hire in Histoire de Vacad6mie r. des sciences, ann6e 1708 (Paris, 1730), "Memoires," p. 57. 7 A. Parent in op. cit., ann<5e 1712 (Paris, 1731), "M6moires," p. 98. 8 Bomic in op. cit., p. 213. 9 Saulmon in op. cit., p. 283. 10 Emanuel Swcdberg, Daedalus Hyperboreus (Upsala, 1716). 11 T. F. Lagny in Histoire de I'academie r. des sciences, ann6e 1719 (Paris, 1721), "Memoires," p. 139. 12 Dominique de Senes in op. cit., p. 363. 13 De Louville in op. cit., ami6e 1724 (Paris, 1726), p. 67. 14 Clairaut in op. cit., ann6e 1731 (Paris, 1733), "M6moires," p. 484. 16 Pierre Bougver in op. cit., anne*e 1733 (Paris, 1735), "Me"moires," p. 89. 16 F. Nicole in op. cit., ann6e 1737 (Paris, 1740), "Me*moires," p. 64. 294 A HISTORY OF MATHEMATICAL NOTATIONS Caille, 1 D'Alembert, 2 Vicenti Riccati, 3 and Jean Bernoulli. 4 In the Latin edition of De la Caille's 5 Lectiones four notations are explained, namely, 3.12::2.8, 3:12::2:8, 3:12 = 2:8, 3|12||2|8, but the nota- tion 3: 12:: 2:8 is the one actually adopted. The notation : : : : was commonly used in England and the United States until the beginning of the twentieth century, and even now in those countries has not fully surrendered its place to : = : . As late as 1921 : :: : retains its place in Edwards' Trigonometry* and it occurs in even later publications. The : : : : gained full ascendancy in Spain and Portugal, and in the Latin- American countries. Thus it was used in Madrid by Juan Gerard, 7 in Lisbon by Joao Felix Pereira 8 and Luiz Porfirio da Motta Pegado, 9 in Rio de Janeiro in Brazil by Fran- cisco Miguel Pires 10 and C. B. Ottoni, 11 at Lima in Peru by Maximo Vazquez 12 and Luis Monsante, 13 at Buenos Ayres by Florentino Garcia, 14 at Santiago de Chile by Gabriel Izquierdo, 15 at Bogota in Colombia by Indalecio Li6vano, 16 at Mexico by Juan Maria Balbontin. 17 1 La Caille in op. ciL, annee 1741 (Paris, 1744), p. 256. 2 D'Alembert in op. ciL, ann<5c 1745 (Paris, 1749), p. 367. 3 Vincent! Riccati, Opusculorum ad res physicas et mathcmaticas pertinentium. Tomus primus (Bologna, 1757), p. 5. 4 Jean Bernoulli in Nouveaux memoires de Vacadernie r. des sciences et belles- lettres, ann6c 1771 (Berlin, 1773), p. 286. 6 N. L. de la Caille, Lectiones elementares malhematicae , ... in Latinum tra- ductae et ad editionem Parisinam anni MDCCL1X denuo cxactae a C [arolo] S [cherffer] e S. J. (Vienna, 1762), p. 76. 6 R. W. K. Edwards, An Elementary Text-Book of Trigonometry (new ed.; London, 1921), p. 152. 7 Juan Gerard, Presbitero, Tratado completo de aritmetica (Madrid, 1798), p. 69. 8 J. F. Pereira, Rudimentos de arithmetica (QuartaEdigao; Lisbon, 1863), p. 129. 9 Luiz Porfirio da Motta Pegado, Tratado elementar de arithmetica (Secunda edigao; Lisbon, 1875), p. 235. 10 Francisco Miguel Pires, Tratado de Trigonometria Espherica (Rio de Janeiro, 1866), p. 8. 11 C. B. Ottoni, Elementos de geometria e trigonometria reclilinea (4th ed.; Rio de Janeiro, 1874), "Trigon.," p. 36. 12 Maximo Vazquez, Aritmetica practica (7th cd.; Lima, 1875), p. 130. 13 Luis Monsante, Lecciones de aritmetica demostrada (7th ed.; Lima, 1872), p. 171. 14 Florentino Garcia, El aritmetica Argentino (5th ed.; Buenos Aires, 1871), p. 41; first edition, 1833. 16 Gabriel Izquierdo, Tratado de aritmetica (Santiago, 1859), p. 157. 16 Indalecio LieVano, Tratado de aritmetica (2d ed.; Bogota, 1872), p. 148. 17 Juan Maria Balbontin, Tratado elemental de aritmetica (Mexico, 1878), p. 96. PROPORTION 295 259. The notation of Leibniz. In the second half of the eighteenth century this notation, A:B::C:D, had gained complete ascendancy over A.B::C.D in nearly all parts of Continental Europe, but at that very time it itself encountered a serious rival in the superior Leibnizian notation, A:B = C:D. If a proportion expresses the equality of ratios, why should the regular accepted equality sign not be thus extended in its application? This extension of the sign of equality = to writing proportions had already been made by Stampioen ( 256). Leibniz introduced the colon (:) for ratio and for division in the Ada eruditorum of 1684, page 470 ( 537). In 1693 Leibniz expressed his disapproval of the use of special symbols for ratio and proportion, for the simple reason that the signs for division and equality are quite sufficient. He 1 says: "Many indicate by a-r-b^rc + d that the ratios a to b and c to d are alike. But I have always disap- proved of the fact that special signs are used in ratio and proportion, on the ground that for ratio the sign of division suffices and likewise for proportion the sign of equality suffices. Accordingly, I write the ratio a to & thus: a:b or j- just as is done in dividing a by b. I desig- nate proportion, or the equality of two ratios by the equality of the two divisions or fractions. Thus when I express that the ratio a to 6 CL C is the same as that of c to d, it is sufficient to write a:b = c:d or r =-3 ." 6 a Cogent as these reasons are, more than a century passed be- fore his symbolism for ratio and proportion triumphed over its rivals. Leibniz 's notation, a:b = c:d, is used in the Ada eruditorum of 1708, page 271. In that volume (p. 271) is laid the editorial policy that in algebra the Leibnizian symbols shall be used in the Ada. We quote the following relating to division and proportion (197): "We shall designate division by two dots, unless circumstance should prompt adherence to the common practice. Thus, we shall have a:6 = r. Hence with us there will be no need of special symbols for denoting proportion. For instance, if a is to b as c is to d, we have a:b = c:d." The earliest influential textbook writer who adopted Leibniz' notation was Christian Wolf. As previously seen ( 255) he sometimes 1 G. W. Leibniz, Matheseos universalis pars prior, de Terrainis incomplexis, No. 16; reprinted in Gesammelte Werke (C. I. Gerhardt), 3. Folge, II 3 , Band VII (Halle, 1863), p. 56. 296 A HISTORY OF MATHEMATICAL NOTATIONS wrote a.b^c.d. In 1710 1 he used both 3.12::5.20 and 3:12 = 5:20, but from 1713 2 on, the Leibnizian notation is used exclusively. One of the early appearances of a : b = c : d in France is in Clairaut's algebra 3 and in Saverien's dictionary, 4 where Saverien argues that the equality of ratios is best indicated by = and that :: is superfluous. It is found in the publications of the Paris Academy for the year 1765, 5 in connection with Euler who as early as 1727 had used it in the com- mentaries of the Petrograd Academy. Benjamin Panchaud brought out a text in Switzerland in 1743, 6 using : = :. In the Netherlands 7 it appeared in 1763 and again in 1775. 8 A mixture of Oughtred's symbol for ratio and the = is seen in Pieter Venema 9 who writes . = . In Vienna, Paulus Mako 10 used Leibniz' notation both for geo- metric and arithmetic proportion. The Italian Petro Giannini 11 used : = : for geometric proportion, as does also Paul Frisi. 12 The first volume of Ada Helvetia 13 gives this symbolism. In Ireland, Joseph Fenn 14 used it about 1770. A French edition of Thomas Simpson's geometry 15 uses : = : . Nicolas Fuss 16 employed it in St. Petersburgh. In England, 1 Chr. Wolf, Anfangsgninde alter mathematischen Wissenschaften (Magdeburg, 1710), Vol. I, p. 65. See J. Tropfke, Geschichte der EUmentar-Mathematik, Vol. Ill (2d ed.; Berlin and Leipzig, 1922), p. 14. 2 Chr. Wolf, Elementa malhescos universae, Vol. I (Halle, 1713), p. 31. 3 A, C. Clairaut, Siemens d'algebre (Paris, 1746), p. 21. 4 A. Saverien, Diclionnaire universel de mathematique et physique (Paris, 1753), arts. "Raisons semblables," "Caractere." 5 Histoire de V academic r. des sciences, anne*e 1765 (Paris, 1768), p. 563; Commentarii academiae scientiarum .... ad annum 1727 (Petropoli, 1728), p. 14. 6 Benjamin Panchaud, Entretiens on lemons mathtmatiques (Lausanne, Geneve, 1743), p. 226. 7 A. R. Maudvit, Inleiding tot de Keegel-Sneeden (Shaage, 1763). 8 J. A. Fas, Inleiding tot de Kennisse en het Gebruyk der Oneindig Kleinen (Ley- den, 1775), p. 80. 9 Pieter Venema, Algebra ofte Stel-Konst, Vierde Druk (Amsterdam, 1768), p. 118. 10 Pavlvs Mako, Compendiaria matheseos institutio (editio altera; Vindobonae, 1766), p. 169, 170. 11 Petro Giannini, Opuscola mathemaiica (Parma, 1773), p. 74. 12 Paulli Frisii Operum, Tomus Secundus (Milan, 1783), p. 284. 18 Ada Helvetica, physico-mathematico-Botanico-Medica, Vol. I (Basel, 1751), p. 87. 14 Joseph Fenn, The Complete Accountant (Dublin, [n.d.]), p. 105, 128. 15 Thomas Simpson, Element de geometric (Paris, 1766). 16 Nicolas Fuss, Lemons de geomttrie (St. PStersbourg, 1798), p. 112. EQUALITY 297 John Cole 1 adopted it in 1812, but a century passed after this date before it became popular there. The Leibnizian notation was generally adopted in Europe during the nineteenth century. In the United States the notation : :: : was the prevailing one during the nineteenth century. The Leibnizian signs appeared only in a few works, such as the geometries of William Chauvenet 2 and Benjamin Peirce. 3 It is in the twentieth century that the notation : = : came to be generally adopted in the United States. A special symbol for variation sometimes encountered in English and American texts is oc , introduced by Emerson. 4 "To the Common Algebraic Characters already received I add this oc , which signifies a BC general Proportion; thus, -4.ee--, signifies that A is in a constant RC 1 ratio to -jr- ." The sign was adopted by Chrystal, 5 Castle, 6 and others. SIGNS OP EQUALITY 260. Early symbols. A symbol signifying "it gives" and ranking practically as a mark for equality is found in the linear equation of the Egyptian Ahmcs papyrus ( 23, Fig. 7). We have seen ( 103) that Diophantus had a regular sign for equality, that the contraction pha answered that purpose in the Bakhshal! arithmetic ( 109), that the Arab al-Qalasadi used a sign ( 124), that the dash was used for the expression of equality by Regiomontanus ( 126), Pacioli ( 138), and that sometimes Cardan ( 140) left a blank space where we would place a sign of equality. 261. Recorders sign of equality. In the printed books before Recordc, equality was usually expressed rhetorically by such words as aequales, aequantur, esgale, faciunt, ghelijck, or gleich, and some- times by the abbreviated form aeq. Prominent among the authors expressing equality in some such manner are Kepler, Galileo, Torri- celli, Cavalieri, Pascal, Napier, Briggs, Gregory St. Vincent, Tacquet, and Fermat. Thus, about one hundred years after Recorde, some of 1 John Cole, Stereogoniometry (London, 1812), p. 44, 265. 2 William Chauvenet, Treatise on Elementary Geometry (Philadelphia, 1872), p. 69. 3 Benjamin Peirce, Elementary Treatise on Plane and Solid Geometry (Boston, 1873), p. xvi. 4 W. Emerson, Doctrine of Fluxions (3d ed.; London, 1768), p. 4. s G. Chrystal, Algebra, Part I, p. 275. 6 Frank Castle, Practical Mathematics for Beginners (London, 1905), p, 317. 298 A HISTORY OF MATHEMATICAL NOTATIONS the most noted mathematicians used no symbol whatever for the expression of equality. This is the more surprising if we remember that about a century before Recorde, Regiomontanus ( 126) in his correspondence had sometimes used for equality a horizontal dash , that the dash had been employed also by Pacioli ( 138) and Ghaligai (139). Equally surprising is the fact that apparently about the time of Recorde a mathematician at Bologna should independently origi- nate the same symbol (Fig. 53) and use it in his manuscripts. Recorders = , after its ddbut in 1557, did not again appear in print until 1618, or sixty-one years later. That some writers used symbols in their private manuscripts which they did not exhibit in their printed books is evident, not only from the practice of Regio- montanus, but also from that of John Napier who used Recorders = in an algebraic manuscript which he did not publish and which was first printed in 1839. 1 In 1618 we find the = in an anonymous Appen- dix (very probably due to Oughtred) printed in Edward Wright's English translation of Napier's famous Descriptio. But it was in 1631 that it received more than general recognition in England by being adopted as the symbol for equality in three influential works, Thomas Harriot's Artis analyticae praxis, William Oughtred's Claris mathematicae, and Richard Norwood's Trigonometria. 262. Different meanings of =. As a source of real danger to Recorde's sign was the confusion of symbols which was threatened on the European Continent by the use of = to designate relations other than that of equality. In 1591 Francis Vieta in his In artem analyticen isagoge used = to designate arithmetical difference ( 177). This designation was adopted by Girard ( 164), by Sieur de Var-Lezard 2 in a translation of Vieta's Isagoge from the Latin into French, De Graaf, 3 and by Franciscus a Schooten 4 in his edition of Descartes' Geometrie. Descartes 5 in 1638 used = to designate plus ou mains, i.e., . Another complication arose from the employment of = by Johann 1 Johannis Napier, De Arte Logistica (Edinburgh, 1839), p. 160. 2 1. L. Sieur de Var-Lezard, Introduction en I'art analytic ov nouvelle algebre de Francois Viete (Paris, 1630), p. 36. 8 Abraham de Graaf, De beginselen van de Algebra of Stelkonst (Amsterdam, 1672), p. 26. * Renati Descartes, Geometria (ed. Franc, a Schooten; Francofvrti al Moenvm, 1695), p. 395. * (Euvres de Descartes (eU Adam et Tannery), Vol. II (Paris, 1898), p. 314, 426. EQUALITY 299 Caramuel 1 as the separatrix in decimal fractions; with him 102 = 857 meant our 102.857. As late as 1706 G. H. Paricius 2 used the signs = , :, and as general signs to separate numbers occurring in the process of solving arithmetical problems. The confusion of algebraic language was further increased when Dulaurens 3 and Reyher 4 desig- nated parallel lines by = . Thus the symbol = acquired five different meanings among different continental writers. For this reason it was in danger of being discarded altogether in favor of some symbol which did not labor under such a handicap. 263. Competing symbols. A still greater source of danger to our = arose from competing symbols. Pretenders sprang up early on both the Continent and in England. In 1559, or two years after the appearance of Recorders algebra, the French monk, J. Buteo, 6 pub- lished his Logistica in which there appear equations like "LA, -J-B, %C [14" and "3A .3#. 15C[120," which in modern notation are x+$y+$z = 14 and 3x+3?/+ 15-2 = 120. Buteo's [ functions as a sign of equality. In 1571, a German writer, Wilhelm Holzmann, better known under the name of Xylander, brought out an edition of Diophantus' Arith- metica* in which two parallel vertical lines || were used for equality. He gives no clue to the origin of the symbol. Moritz Cantor 7 suggests that perhaps the Greek word iaoi ("equal") was abbreviated in the manuscript used by Xylander, by the writing of only the two letters u. Weight is given to this suggestion in a Parisian manuscript on Diophantus where a single i denoted equality. 8 In 1613, the Italian writer Giovanni Camillo Glorioso used Xylander's two vertical lines for equality. 9 It was used again by the Cardinal Michaelangelo Ricci. 10 This character was adopted by a few Dutch and French 1 Joannis Caramuelis, Mathesis Biceps veins et nova (1670), p. 7. 2 Georg Heinrich Paricius, Praxis arithmelices (Regensburg, 1706). Quoted by M. Sterner, Geschichte der Rechenkunst (Munchen und Leipzig, 1891), p. 348. 3 Franc. ois Dulaurens, Specimina mathemalica (Paris, 1667). 4 Samuel Reyher, Euclides (Kiei, 1698). 8 J. Buteo, Logistica (Leyden, 1559), p. 190, 191. See J. Tropfke, op. cit., Vol. Ill (2d ed.; Leipzig, 1922), p. 136. 6 See Nesselmann, Algebra der Griechen (1842), p. 279. 7 M. Cantor, Vorlesungen uber Geschichte der Mathematik, Vol. II (2d ed.; Leipzig, 1913), p. 552. 8 M. Cantor, op. cit., Vol. I (3d ed.; 1907), p. 472. 9 Joannis Camillo Gloriosi t Ad theorema geometricvm (Venetiis, 1613), p. 26. 10 Michaelis Angeli Riccii, Exercitatio geometrica de maximis el minimis (Lon- dini, 1668), p. 9. 300 * A HISTORY OF MATHEMATICAL NOTATIONS mathematicians during the hundred years that followed, especially in the writing of proportion. Thus, R. Descartes, 1 in his Opuscules de 1619-1621 , made the statement, "ex progressione 1 1 2 1 1 4 1 8 1 1 16 1 32 1| habentur numeri perfecti 6, 28, 496." Pierre de Carcavi, of Lyons, in a letter to Descartes (Sept. 24, 1649), writes the equation "+1296 3060a+2664a 2 -1115aM-239a 4 -25a 5 +a 6 1| 0," where "la lettre a est Pinconnue en la rnaniere de Monsieur Vieta" and || is the sign of equality. 2 De Monconys 3 used it in 1666; De Sluse 4 in 1668 writes our be=a? in this manner "be \\ aa." De la Hire (254) in 1701 wrote the proportion a:b = x 2 :ab thus: "a|6||o;x|ab." This symbolism is adopted by the Dutch Abraham de Graaf 5 in 1703, by the Frenchman Parent 6 in 1713, and by certain other writers in the Journal des S$avans. 7 Though used by occasional writers for more than a century, this mark || never gave promise of becoming a universal symbol for equality. A single vertical line was used for equality by S. Ileyher in 1698. With him, "A\B" meant A = #. He attributes 8 this notation to the Dutch orientalist and astronomer Jacob Golius, saying: "Espe- cially indebted am I to Mr. Golio for the clear algebraic mode of dem- onstration with the sign of equality, namely the rectilinear stroke standing vertically between two magnitudes of equal measure." In England it was Leonard and Thomas Digges, father and son, who introduced new symbols, including a line complex X for equality (Fig. 78).' The greatest oddity was produced by H6rigone in his Cursus mathematicus (Paris, 1644; 1st ed., 1634). It was the symbol "2|2." Based on the same idea is his "3|2" for "greater than," and his "2|3" for "less than." Thus, a?+ab = b 2 is indicated in his symbolism by 1 (Euvrcs de Descartes, Vol. X (1908), p. 241. 2 Op. cit., Vol. V (1903), p. 418. 3 Journal des voyages de Monsieur de Monconys (Troisie'me partie ; Lyon, 166G), p. 2. Quoted by Henry in Revue archeologique (N.S.), Vol. XXXVII (1879), p. 333. 4 Renati Francisci Slusii Mesolabum, Leodii Eburonum (1668), p. 51. 5 Abraham de Graaf, De Vervulling van de Geometria en Algebra (Amsterdam, 1708), p. 97. 8 A. Parent, Essais el recherches de mathematique et de physique (Paris, 1713), p. 224. 7 Journal des Sc.avans (Amsterdam, for 1713), p. 140; ibid, (for 1715), p. 537; and other years. 8 Samuel Ileyher, op. cit., Vorrede. 9 Thomas Digges, Stralioticos (1590), p. 35. EQUALITY 301 "a2+6a2|262." Though clever and curious, this notation did not appeal. In some cases Hrigone used also LJ to express equality. If this sign is turned over, from top to bottom, we have the one used by F. Dulaurens 1 in 1667, namely, n ; with Dulaurens P signifies "majus," H signifies "minus"; Leibniz, in some of his correspondence and unpublished papers, used 2 n and also 3 = ; on one occasion he used the Cartesian 4 for identity. But in papers which he printed, only the sign = occurs for equality. Different yet was the equality sign 3 used by J. V. Andrea 5 in 1614. The substitutes advanced by Xylander, Andrea, the two Digges, Dulaurens, and Herigone at no time seriously threatened to bring about the rejection of Recorders symbol. The real competitor was the mark , prominently introduced by Ren6 Descartes in his Geometric (Leyden, 1637), though first used by him at an earlier date. 6 264. Descartes 1 sign of equality. It has been stated that the sign was suggested by the appearance of the combined ae in the word aequalis, meaning "equal." The symbol has been described by Cantor 7 as the union of the two letters ae. Better, perhaps, is the description given by Wieleitner 8 who calls it a union of oe reversed; his minute examination of the symbol as it occurs in the 1637 edition of the Geometric revealed that not all of the parts of the letter e in the combination oe are retained, that a more accurate way of describing that symbol is to say that it is made up of two letters o, that is, oo pressed against each other and the left part of the first excised. In some of the later appearances of the symbol, as given, for example, by van Schooten in 1659, the letter e in oe, reversed, remains intact. We incline to the opinion that Descartes' symbol for equality, as it appears in his Geometric of 1637, is simply the astronomical symbol 1 F. Dulaurens, Specimina mathemalica (Paris, 1667). 2 C. I. Gerhardt, Leibnizens mathematische Schriften, Vol. I, p. 100, 101, 155, 163, etc. 3 Op. dt., Vol. I, p. 29, 49, 115, etc. 4 Op. ciL, Vol. V, p. 150. 5 Joannis Valentini Andreae, Collectaneorum Mathematicorum decades XI (Tubirigae, 1614). Taken from P. Treutlein, "Die deutsche Coss," Abhandlungen zur Geschichte der Mathematik, Vol. II (1879), p. 60. * (Entires de Descartes (ed. Ch. Adam et P. Tannery), Vol. X (Paris, 1908), p. 292, 299. 7 M. Cantor, op. cit., Vol. II (2d ed., 1913), p. 794. 8 II. Wieleitner in Zeitschr. fur math. u. naiuriviss. Unterricht, Vol. XL VII (1916), p. 414. 302 A HISTORY OF MATHEMATICAL NOTATIONS for Taurus, placed sideways, with the opening turned to the left. This symbol occurs regularly in astronomical works and was there- fore available in some of the printing offices. Descartes does not mention Recorde's notation; his Geometrie is void of all bibliographical and historical references. But we know that he had seen Harriot's Praxis, where the symbol is employed regularly. In fact, Descartes himself 1 used the sign = for equality in a letter of 1640, where he wrote "1C-6# = 40" for 0^-60; = 40. Descartes does not give any reason for advancing his new symbol *> . We surmise that Vieta's, Girard's, and De Var-Lezard's use of = to denote arith- metical "difference" operated against his adoption of Recorde's sign. Several forces conspired to add momentum to Descartes' symbol . In the first place, the Geometrie, in which it first appeared in print, came to be recognized as a work of genius, giving to the world analytic geometry, and therefore challenging the attention of mathematicians. In the second place, in this book Descartes had perfected the expo- nential notation, a n (n, a positive integer), which in itself marked a tremendous advance in symbolic algebra; Descartes' was likely to follow in the wake of the exponential notation. The was used by F. Debeaune 2 as early as October 10, 1638, in a letter to Roberval. As Descartes had lived in Holland several years before the appear- ance of his Geometrie, it is not surprising that Dutch writers should be the first to adopt widely the new notation. Van Schooten used the Cartesian sign of equality in 1646. 3 He used it again in his translation of Descartes' Geometrie into Latin (1649), and also in the editions of 1659 and 1695. In 1657 van Schooten employed it in a third publica- tion. 4 Still more influential was Christiaan Huygens 5 who used as early as 1646 and in his subsequent writings. He persisted in this usage, notwithstanding his familiarity with Recorde's symbol through the letters he received from Wallis and Brounckcr, in which it occurs many times. 6 The Descartian sign occurs in the writings of Hudde and De Witt, printed in van Schooten's 1659 and later editions of Descartes' Geometrie. Thus, in Holland, the symbol was adopted by 1 (Euvres dc Descartes, Vol. Ill (1899), p. 190. 2 Ibid., Vol. V (1903), p. 519. 3 Francisci i\ Schooten, DC organica conicarum sectionum (Leyden, 1646), p. 91. 4 Francisci & Schooten, Exercitationvm malhematicarum liber primus (Lcyden, 1657), p. 251. 6 (Euvres completes de Christiaan Huygens , Tome I (La Haye, 1888), p. 26, 526. 6 Op. cit., Tome II, p. 296, 519; Tome IV, p. 47, 88. EQUALITY 303 the most influential mathematicians of the seventeenth century. It worked its way into more elementary textbooks. Jean Prestet 1 adopted it in his Nouveaux Elemens, published at Paris in 1689. This fact is the more remarkable, as in 1675 he 2 had used the sign = . It seems to indicate that soon after 1675 the sign was gaining over = in France. Ozanam used in his Didionaire mathematique (Amster- dam, 1691), though in other books of about the same period he used ^, as we see later. The Cartesian sign occurs in a French text by Bernard Lamy. 3 In 1659 Descartes' equality symbol invaded England, appearing in the Latin passages of Samuel Foster's Miscellanies. Many of the Latin passages in that volume are given also in English translation. In the English version the sign = is used. Another London publica- tion employing Descartes' sign of equality was the Latin translation of the algebra of the Swiss Johann Alexander. 4 Michael Rolle uses in his Traite d'algebre of 1690, but changes to = in 1709. 5 In Hol- land, Descartes' equality sign was adopted in 1660 by Kinckhvysen, 6 in 1694 by De Graaf, 7 except in writing proportions, when he uses =. Bernard Nieuwentiit uses Descartes' symbol in his Considerations of 1694 and 1696, but preferred = in his Analysis infinitorum of 1695. De la Hire 8 in 1701 used the Descartian character, as did also Jacob Bernoulli in his Ars Conjectandi (Basel, 1713). Descartes' sign of equality was widely used in France and Holland during the latter part of the seventeenth and the early part of the eighteenth centuries, but it never attained a substantial foothold in other countries. 265. Variations in the form of Descartes 1 symbol. Certain varia- tions of Descartes' symbol of equality, which appeared in a few texts, are probably due to the particular kind of symbols available or irn- provisable in certain printing establishments. Thus Johaan Cara- 1 Jean Prestet, Nouveaux Siemens des mathemaliques, Vol. I (Paris, 1689), p. 261. 2 J. P. [restet] Siemens des mathtmaliques (Paris, 1675), p. 10. 3 Bernard Lamy, Elemens des mathemaliques (3d ed.; Amsterdam, 1692), p. 93. 4 Synopsis Algebraica, Opus posthumum Johannis Alexandri, Bernalis-Helvetii. In usurn scholae malhematicae apud Hospitium-Christi Londinense (Londirii, 1693), p. 2. 5 Mem. de Vacademie royale des sciences, anne*e 1709 (Paris), p. 321. 6 Gerard Kinckhvysen, De Grondt der Meet-Konst (Te Haerlem, 1660), p. 4. 7 Abraham de Graaf, De Geheele Mathesis of Wiskonst (Amsterdam, 1694), p. 45. 8 De la Hire, Nouveaux tttmens des sections coniques (Paris, 1701), p. 184. 304 A HISTORY OF MATHEMATICAL NOTATIONS muel 1 in 1670 employed the symbol 7E; the 1679 edition of Format's 2 works gives oo in the treatise Ad locos pianos et solidos isagoge, but in Fermat's original manuscripts this character is not found. 8 On the margins of the pages of the 1679 edition occur also expressions of which "DA {BE" is an example, where DA=BE. J. Ozanam 4 em- ploys V^N in 1682 and again in 1693; he refers to as used to mark equality, "mais nous le changerons en celuy-cy, oo ; que nous semble plus propre, et plus naturel." Andreas Spole 5 said in 1692: u ~ vel = est nota aequalitates." Wolff 6 gives the Cartesian symbol inverted, thus a . 266. Struggle for supremacy. In the seventeenth century, Recorders = gained complete ascendancy in England. We have seen its great rival in only two books printed in England. After Harriot and Oughtred, Recorders symbol was used by John Wallis, Isaac Barrow, and Isaac Newton. No doubt these great names helped the symbol on its way into Europe. On the European Continent the sign = made no substantial headway until 1650 or 1660, or about a hundred years after the appear- ance of Recorders algebra. When it did acquire a foothold there, it experienced sharp competition with other symbols for half a century before it fully established itself. The beginning of the eighteenth century may be designated roughly as the time when all competition of other symbols practically ceased. Descartes himself used = in a letter of September 30, 1640, to Mersenne. A Dutch algebra of 1639 and a tract of 1640, both by J. Stampioen, 7 and the Teutsche Algebra of the Swiss Johann Heinrich Rahn (1659), are the first continental textbooks that we have seen which use the symbol. Rahn says, p. 18: "Bey disem anlaasz habe ich das namhafte gleichzeichen = zum ersten gebraucht, bedeutet ist gleich, 2a = 4 heisset 2a ist gleich 4." It was used by Bernhard Frenicle de Bessy, of magic-squares fame, in a 1 J. Caramuel, op. cit., p. 122. 2 Varia opera mathematica D. Petri de Fermai (Tolosae, 1679), p. 3, 4, 5. 8 (Euvres de Fermat (ed. P. Tannery et C. Henry), Vol. I (Paris, 1891), p. 91. 4 Journal des Sgavans (de Tan 1682), p. 160; Jacques Ozanam, Cours de Mathe- matiques, Tome I (Paris, 1692), p. 27; also Tome III (Paris, 1693), p. 241. 6 Andreas Spole, Arithmetica vulgaris et specioza (Upsaliae, 1692), p. 16. See G. Enestrom in U Intermediaire des mathematiciens, Tome IV (1897), p. 60. 6 Christian Wolff, Maihemalisches Lexicon (Leipzig, 1716), "Signa," p. 1264. 7 Johan Stampioen d'Jonghe, Algebra ofte Nieuwe Stel-Regel ('s Graven-Hage, 1639); J. Stampioenii Wisk-Konstich ende Reden-maetich Bewijs (s'Graven-IIage, 1640). EQUALITY 305 letter 1 to John Wallis of December 20, 1661, and by Huips 2 in the same year. Leibniz, who had read Barrow's Euclid of 1655, adopted the Recordean symbol, in his De arte combinatoria of 1666 ( 545), but then abandoned it for nearly twenty years. The earliest textbook brought out in Paris that we have seen using this sign is that of Arnauld 3 in 1667; the earliest in Leyden is that of C. F. M. Dechales 4 in 1674. The sign = was used by Prestet, 5 Abbe Catelan and Tschirnhaus, 6 Hoste, 7 Ozanam, 8 Nieuwentijt, 9 Wcigel, 10 De Lagny, 11 Carre, 12 L'Hospi- tal, 13 Polynier, 14 Guisnee, 15 and Reyneau. 16 This list constitutes an imposing array of names, yet the majority of writers of the seventeenth century on the Continent either used Descartes' notation for equality or none at all. 267. With the opening of the eighteenth century the sign = gained rapidly; James Bernoulli's Ars Conjectandi (1713), a post- humous publication, stands alone among mathematical works of prominence of that late date, using . The dominating mathematical advance of the time was the invention of the differential and integral calculus. The fact that both Newton and Leibniz used Recorde's symbol led to its general adoption. Had Leibniz favored Descartes' I (Euvres completes des Christiaan-Huygens (Lallayc), Tome IV (1891), p. 45. 2 Frans van der Huips, Algebra ofte een Noodige (Amsterdam, 1601), p. 178. Reference supplied by L. C. Karpinski. 3 Antoine Arnauld, Nouveaux Siemens de Geometric (Paris, 1667; 2d ed., 1683). 4 C. F. Dechales, Cvrsvs sev Mvndvs Mathematics, Tomvs tertivs (Lvgdvni, 1674), p. 666; Editio altera, 1690. 5 J. P[restet], op. ciL (Paris, 1675), p. 10. 6 Ada eruditorum (anno 1682), p. 87, 393. 7 P. Hoste, Recueil des traites de mathematiques, Tome III (Paris, 1692), p. 93. 8 Jacques Ozanam, op. cit., Tome I (nouvelle 6d.; Paris, 1692), p. 27. In various publications between the dates 1682 and 1693 Ozanam used as equality signs / ', , and =. 9 Bernard Nieuwentijt, Analysis infinilorum. 10 Erhardi Wvigelii Philosophia mathematica (Jcnae, 1693), p. 135. II Thomas F. de Lagny, Nouveaux eUrnens d'arithrnetique, et d'algebre (Paris, 1697), p. 232. 12 Louis Carre*, Methode pour la mesure des surfaces (Paris, 1700), p. 4. 13 Marquis de I'Hospital, Analyse des Infmiment Petits (Paris, 1696, 1715). 14 Pierre Polynier, EUmens des Mathematiques (Paris, 1704), p. 3. 16 Guisnee, Application de Valgebre a la geometric (Paris, 1705). 16 Charles Reyneau, Analyse demontree, Tome I (1708). 306 A HISTORY OF MATHEMATICAL NOTATIONS , then Germany and the rest of Europe would probably have joined France and the Netherlands in the use of it, and Recorded symbol would probably have been superseded in England by that of Descartes at the time when the calculus notation of Leibniz displaced that of Newton in England. The final victory of = over seems mainly due to the influence of Leibniz during the critical period at the close of the seventeenth century. The sign of equality = ranks among the very few mathematical symbols that have met with universal adoption. Recorde proposed no other algebraic symbol; but this one was so admirably chosen that it survived all competitors. Such universality stands out the more prominently when we remember that at the present time there is still considerable diversity of usage in the group of symbols for the differ- ential and integral calculus, for trigonometry, vector analysis, in fact, for every branch of mathematics. The difficulty of securing uniformity of notation is further illus- trated by the performance of Peter van Musschenbroek, 1 of Leyden, an eighteenth-century author of a two-volume text on physics, widely known in its day. In some places he uses = for equality and in others for ratio; letting S. s. be distances, and T. t .times, he says: "Erit S. s. :: T. t. exprimunt hoc Mathematici scribendo, est S= T. sive Spatium est uti tempus, nam signum = non exprimit aequalitatem, sed ratio- nem." In writing proportions, the ratio is indicated sometimes by a dot, and sometimes by a comma. In 1754, Musschenbroek had used for equality. 2 268. Variations in the form of Recorde's symbol. There has been considerable diversity in the form of the sign of equality. Recorde drew the two lines very long (Fig. 71) and close to each other, . This form is found in Thomas Harriot's algebra (1631), and occa- sionally in later works, as, for instance, in a paper of De Lagny 3 and in Schwab's edition of Euclid's Data. 4 Other writers draw the two lines very short, as does Weigel 5 in 1693. At Upsala, Emanuel van Musschenbroek, Introdttctio ad philosophiam naluralem, Vol. I (Leyden, 1762), p. 75, 126. 2 Petri van Musschenbroek, Dissertaiio physica experimentalis de magnete (Vienna), p. 239. 8 De Lagny in Memvires de I'acadtmie r. d. sciences (depuis 1666 jusqu'a 1699), Vol. II (Paris, 1733), p. 4. 4 Johann Christoph Schwab, Eudida Data (Stuttgart, 1780), p. 7. Erhardi Weigeli PhUosophia mathematica (Jena, 1693), p. 181. EQUALITY 307 Swedenborg 1 makes them very short and slanting upward, thus //. At times one encounters lines of moderate length, drawn far apart z , as in an article by Nicole 2 and in other articles, in the Journal des S$avans. Frequently the type used in printing the symbol is the figurfc 1, placed horizontally, thus 3 ^ or 4 In an American arithmetic 5 occurs, "1+6, = 7, X6 = 42, 4-2 = 21." Wolfgang Bolyai 6 in 1832 uses =z to signify absolute equality; 21, equality in content; A(~B or B = )A,to signify that each value of A is equal to some value of B; A( = )B, that each of the values of A is equal to some value of B, and vice versa. To mark the equality of vectors, Bellavitis 7 used in 1832 and later the sign ==. Some recent authors have found it expedient to assign = a more general meaning. For example, Stolz and Grneiner 8 in their theoretical arithmetic write a o 6 = c and read it "a mit 6 ist c," the = signifying "is explained by" or "is associated with." The small circle placed between a and b means, in general, any relation or Verknupfung. De Morgan 9 used in one of his articles on logarithmic theory a double sign of equality = = in expressions like (be 0l/ ~ 1 ) x = = ne lf ^ / ~ 1 J where ft and v are angles made by 6 and n, respectively, with the initial line. He uses this double sign to indicate "that every symbol shall express not merely the length and direction of a line, but also the quantity of revolution by which a line, setting out from the unit line, is supposed to attain that direction." 1 Emanuel Swedberg, Daedalus Hyperboreus (Upsala, 1716), p. 39. See fac- simile reproduction in Kungliga Vctenskaps Societelens i Upsala Tvdhundradrsminne (Upsala, 1910). 2 Francois Nicole in Journal des Sgavans, Vol. LXXXIV (Amsterdam, 1728), p. 293. See also anne'e 1690 (Amsterdam, 1691), p. 468; ann6e 1693 (Amsterdam, 1694), p. 632. 3 James Gregory, Geometria Pars Vniversalis (Padua, 1668); Emanuel Swed- berg, op. cit., p. 43. 4 H. Vitalis, Lexicon mathematicum (Rome, 1690), art. "Algebra." 5 The Columbian Arithmetician, "by an American" (Haverhill [Mass.], 1811), p. 149. Wolfgang! Bolyai de Bolya, Tentamen (2ded.), Tom. I (Budapestini, 1897), p. xi. 7 Guisto Bellavitis in Annali del R. Lomb.-Ven. (1832), Tom. II, p. 250-53. 8 0. Stolz und J. A. Gmeiner, Theoretische Arithmetik (Leipzig), Vol. I (2d ed.; 1911), p. 7. 9 A. de Morgan, Trans. Cambridge PhUos. Society, Vol. VII (1842), p. 186. 308 A HISTORY OF MATHEMATICAL NOTATIONS 269. Variations in the manner of using it. A rather unusual use of equality signs is found in a work of Deidier 1 in 1740, viz., ^ . ^j = , 2+2+2 = 6"" 2 ; 4,4, 4, = 12~3 + 12* H. Vitalis 2 uses a modified symbol: "Nota == significat repctitam aequationem . . . . vt 10 6. "f~4^8~|~2." A discrimination between = and is made by Gallimard 3 and a few other writers; " = , est egale a; qui signifie tout simplcmcnt, egal a , ou , qui est egal a." A curious use, in the same expressions, of = , the comma, and the word aequalis is found in a Tacquet-Whiston 4 edition of Euclid, where one reads, for example, "erit 8X432 = 3456 aequalis 8X400 = 3200, +8X30 = 240, +8X2 = 16." L. Gustave du Pasquier 5 in discussing general complex numbers employs the sign of double equality = to signify "equal by definition." The relations between the coefficients of the powers of x in a series may be expressed by a formal equality involving the series as a whole, as in where the symbol ^f indicates that the equality is only formal, not arithmetical. 6 270. Nearly equal. Among the many uses made in recent years of the sign ^> is that of "nearly equal to/' as in "e~\"] similarly, e^\ is allowed to stand for "equal or nearly equal to." 7 A. Eucken 8 lets r: stand for the lower limit, as in "J~45.10- 40 (untere Grenze)," where J means a mean moment of inertia. Greenhill 9 denotes approximate 1 L'Abb6 Deidier, La mcsure des surfaces et des solides (Paris, 1740), p. 9. 2 H. Vitalis, loc. cit. 3 J. E. Gallimard, La Science du calcul numerique, Vol. I (Paris, 1751), p. 3. 4 Andrea Tacquet, Elementa Euclidea geometriae [after] Guliclmus Whiston (Amsterdam, 1725), p. 47. 5 Comptes Rendus du Congres International des Mathematicians (Strasbourg, 22-30 Septembre 1920), p. 164. 6 Art. "Algebra" in Encyclopaedia Britannica (1 1th ed., 1910). 7 A. Kratzer in Zeitschrift fur Physik, Vol. XVI (1923), p. 356, 357. 8 A. Eucken in Zeitschrifl der physikalischen Chernie, Band C, p. 159. 9 A. G. Greenhill, Applications of Elliptic Functions (London, 1892), p. 303, 340, 341. COMMON FRACTIONS 309 equality by ^w- An early suggestion due to Fischer 1 was the sign X for "approximately equal to." This and three other symbols were proposed by Boon 2 who designed also four symbols for "greater than but approximately equal to" and four symbols for "less than but approximately equal to." SIGNS OF COMMON FRACTIONS 271. Early forms. In the Egyptian Ahmes papyrus unit fractions were indicated by writing a special mark over the denominator (22, 23). Unit fractions are not infrequently encountered among the Greeks (41), the Hindus and Arabs, in Leonardo of Pisa (122), and in writers of the later Middle Ages in Europe. 3 In the text Trisatika, written by the Hindu Sridhara, one finds examples like the following: "How much money is there when half a kdkini, one-third of this and one-fifth of this are added together? Statement 1 1 1 1 2 3 Answer. Vardtikas 14." This means lX-J + lXiXi+lX!X-i-Xi = 1 \, and since 20 varatikas = 1 kdkini, the answer is 14 varatikas. John of Meurs (early fourteenth century) 4 gives % as the sum of three unit fractions -, -j, and -g^, but writes "\ \ -J-," which is an ascending continued fraction. He employs a slightly different nota- tion for -gV, namely, "i -J - o \ o ." Among Heron of Alexandria and some other Greek writers the numerator of any fraction was written with an accent attached, and was followed by the denominator marked with two accents (41). In some old manuscripts of Diophantus the denominator is placed above the numerator ( 104), and among the Byzantines the denominator is found in the position of a modern exponent ; 5 $ ltt signified according- ly A 1 Ernst Gottfried Fischer, Lehrbuch der Elementar-Mathematik, 4. Theil, Anfangsgrunde der Algebra (Berlin und Leipzig, 1829), p. 147. Reference given by R. C. Archibald in Mathematical Gazette, Vol. VIII (London, 1917), p. 49. 2 C. F. Boon, Mathcinatical Gazette, Vol. VII (London, 1914), p. 48. 3 See G. Encstrom in Bibliotheca mathematica (3d ser.), Vol. XIV (1913-14), p. 269, 270. 4 Vienna Codex 4770, the Quadripartitum numervrum, described by L. C. Karpinski in Bibliotheca mathematics (3d scr.), Vol. XIII (1912-13), p, 109. 5 F. Hultsch, Metrologicorurn scriptorum reliquiae, Vol. I (Leipzig, 1864), p. 173-75. 310 A HISTORY OF MATHEMATICAL NOTATIONS The Hindus wrote the denominator beneath the numerator, but without a separating line ( 106, 109, 113, 235). In the so-called arithmetic of John of Seville, 1 of the twelfth century (?), which is a Latin elaboration of the arithmetic of al- Khowarizmi, as also in a tract of Alnasavi (1030 A.D.), 2 the Indian mode of writing fractions is followed ; in the case of a mixed number, the fractional part appears below the integral part. Alnasavi pur- sues this course consistently 3 by writing a zero when there is no intc- o Jy gral part; for example, he writes ^ thus: i , 272. The fractional line is referred to by the Arabic writer al- Ha$$ar ( 122, 235, Vol. II 422), and was regularly used by Leonardo of Pisa ( 122, 235). The fractional line is absent in a twelfth-century Munich manuscript; 4 it was not used in the thirteenth-century writ- ings of Jordanus Nemorarius, 5 nor in the Gcrnardus algorithmus demonstratus, edited by Joh. Schoner (Niirnberg, 1534), Part II, chapter i. 6 When numerator and denominator of a fraction are letters, Gernardus usually adopted the form ab (a numerator, b denominator), probably for graphic reasons. The fractional line is absent in the Bamberger arithmetic of 1483, but occurs in Widman (1489), and in a fifteenth-century manuscript at Vienna. 7 While the fractional line carne into general use in the sixteenth century, instances of its omis- sion occur as late as the seventeenth century. 273. Among the sixteenth- and seventeenth-century writers omitting the fractional line were Baeza 8 in an arithmetic published at Paris, Dibuadius 9 of Denmark, and Paolo Casati. 10 The line is 1 Boncompagni, Trattati d' aritmetica, VoL II, p. 16-72, 2 H. Suter, Bibliotheca mathematica (3d ser.), Vol. VII (1906-7), p. 113-19. 3 M. Cantor, op. cil. t VoL I (3d ed.), p. 762. 4 Munich MS Clm 13021. See Abhandlungen uber Geschichle der Mathematik, Vol. VIII (1898), p. 12-13, 22-23, and the peculiar mode of operating with frac- tions. 6 Bibliotheca malhematica (3d ser.), VoL XIV, p. 47. 6 Ibid., p. 143. 7 Codex Vindob. 3029, described by E. Rath in Bibliotheca mathematica (3d ser.), VoL XIII (1912-13), p. 19. This manuscript, as well as Widman's arithmetic of 1489, and the anonymous arithmetic printed at Bambcrg in 1483, had as their common source a manuscript known as Algorismus Ratisponensis. 8 Nvmerandi doctrina authore Lodoico Baeza (Lvtctia, 1556), fol. 45. 9 C. Dibvadii in arithmeticam irralionalivm Evclidis (Arnhemii, 1605). 10 Paolo Casati, Fabrica et Vao Del Composso di Proportione (Bologna, 1685) [Privilege, 1662], p. 33, 39, 43, 63, 125. COMMON FRACTIONS 311 usually omitted in the writings of Marin Mersenne 1 of 1644 and 1647. It is frequently but not usually omitted by Tobias Beutel. 2 In the middle of a fourteenth-century manuscript 3 one finds the > I HH notation 3 5 for f, 4 7 for -4- A Latin manuscript, 4 Paris 7377 A, which is a translation from the Arabic of Abu Kamil, contains the fractional line, as in J, but -J-J is a continued fraction and stands for j plus ^ r > whereas $ \ as well as -jj| represent simply -fa. Similarly, Leonardo of Pisa, 5 who drew extensively from the Arabic of Abu Kamil, lets - 8 5 Jj- stand for - f , there being a difference in the order of reading. Leonardo read from right to left, as did the Arabs, while authors of Latin manuscripts of about the fourteenth century read as we do from left to right. In the case of a mixed number, like 3J, Leonardo and the Arabs placed the integer to the right of the fraction. 274. Special symbols for simple fractions of frequent occurrence are found. The Ahmes papyrus has special signs for J and | (22); there existed a hieratic symbol for j ( 18). Diophantus employed special signs for | and | ( 104). A notation to indicate one-half, almost identical with one sometimes used during the Middle Ages in connection with Roman numerals, is found in the fifteenth century with the Arabic numerals. Says Cappelli: "I remark that for the des- ignation of one-half there was used also in connection with the Arabic numerals, in the XV. century, a line between two points, as 4 -~ for 4J, or a small cross to the right of the number in place of an exponent, as 4*, presumably a degeneration of 1/1, for in that century this form was used also, as 7 1/1 for 7 J. Toward the close of the XV. century one finds also often the modern form |." 6 The Roman designation of certain unit fractions are set forth in 58. The peculiar designations employed in the Austrian cask measures are found in 89. In a fif- teenth-century manuscript we find: "Whan pou hayst write pat, for pat pat leues, write such a merke as is here w vpon his hede, pe quych 1 Marin Mersenne, Cogitata Phy&ico-mathemalica (Paris, 1644), "Phaenomena ballistica"; Novarvm observationvm Physico-mathematicarvm y Tomvs III (Paris, 1647), p. 194 ff. 2 Tobias Beutel, Geometrische Gallerie (Leipzig, 1690), p. 222, 224, 236, 239, 240, 242, 243, 246. 3 Bibliothcca mathemdtica (3d ser.), Vol. VII, p. 308-9. L. C. Karpinski in ibid., Vol. XII (1911-12), p. 53, 54. 5 Leonardo of Pisa, Liber abbaci (ed. B. Boncompagni, 1857), p. 447. Note- worthy here is the use of e to designate the absence of a number. A. Cappelli, Lexicon Abbreviaturarum (Leipzig, 1901), p. L. 312 A HISTORY OF MATHEMATICAL NOTATIONS merke schal betoken halfe of pe odde pat was take away"; 1 for ex- ample, half of 241 is 120^. In a mathematical roll written apparently in the south of England at the time of Recorde, or earlier, the char- acter ~ stands for one-half, a dot for one-fourth, and ~ for three- fourths. 2 In some English archives 3 of the sixteenth and seventeenth centuries one finds one-half written in the form ~j . In the earliest arithmetic printed in America, the Arle para aprendar todo el menor del arithmetica of Pedro Paz (Mexico, 1623), the symbol JL is used for | a few times in the early part of the book. This symbol is taken from the Arithmetica practica of the noted Spanish writer, Juan Perez de Moya, 1562 (14th ed., 1784, p. 13), who uses JQ. and also for ^ or media. This may be a convenient place to refer to the origin of the sign % for "per cent/' which has been traced from the study of manuscripts by D. E. Smith. 4 He says that in an Italian manuscript an "unknown writer of about 1425 uses a symbol which, by natural stages, developed into our present %. Instead of writing ' per 100', 'P 100' or T cento/ as had commonly been done before him, he wrote 'Per- 2 ' o o for 'I? 8/ just as the Italians wrote 1, 2, ... and 1, 2, ... for primo, secundo, etc. In the manuscripts which I have examined the evolution is easily traced, the o-* becoming -JJ- about 1650, the original meaning having even then been lost. Of late the 'per' has been dropped; leaving only {j or %." By analogy to %, which is now made up of two zeros, there has been introduced the sign % , having as many zeros as 1,000 and signifying per milled Cantor represents the fraction (100+/>)/100 "by the sign 1, Op, not to be justified mathematically but in practice extremely convenient." 275. The solidus* The ordinary mode of writing fractions r is typographically objectionable as requiring three terraces of type. An effort to remove this objection was the introduction of the solidus, as in a/6, where all three fractional parts occur in the regular line of type. It was recommended by De Morgan in his article on "The Calculus 1 R. Steele, The Earliest Arithmetics in English (London, 1922), p. 17, 19. The p in "pou," "pat," etc., appears to be our modern th. 2 D. E. Smith in American Mathematical Monthly, Vol. XXIX (1922), p. 63. 3 Antiquaries Journal, Vol. VI (London, 1926), p. 272. 4 D. E. Smith, Rara arithmetica (1898), p. 439, 440. 5 Moritz Cantor, Politische Arithmetik (2. Aufl.; Leipzig, 1903), p. 4. 8 The word "solidus" in the time of the Roman emperors meant a gold coin (a "solid" piece of money) ; the sign / comes from the old form of the initial letter s, namely, f, just as is the initial of libra ("pound"), and d of denarius ("penny"). COMMON FRACTIONS 313 of Functions/' published in the Encyclopaedia Metropolitana (1845). But practically that notation occurs earlier in Spanish America. In the Gazetas de Mexico (1784), page 1, Manuel Antonio Valdes used a curved line resembling the sign of integration, thus 1/4, 3/4; Henri Cambuston 1 brought out in 1843, at Monterey, California, a small arithmetic employing a curved line in writing fractions. The straight solidus is employed, in 1852, by the Spaniard Antonio Serra Y Oli- veres. 2 In England, De Morgan's suggestion was adopted by Stokes 3 in 1880. Cayley wrote Stokes, "I think the 'solidus' looks very well indeed . . . . ; it would give you a strong claim to be President of a Society for the Prevention of Cruelty to Printers." The solidus is used frequently by Stolz and Gmeiner. 4 While De Morgan recommended the solidus in 1843, he used a: b in his subsequent works, and as Glaisher remarks, "answers the pur- pose completely and it is free from the objection to -f- viz., that the pen must be twice removed from the paper in the course of writing it." 5 The colon was used frequently by Leibniz in writing fractions ( 543, 552) and sometimes also by Karsten, 6 as in 1:3 = J; the -f- was used sometimes by Cayley. G. Peano adopted the notation b/a whenever it seemed con- venient. 7 Alexander Macfarlane 8 adds that Stokes wished the solidus to take the place of the horizontal bar, and accordingly proposed that the terms immediately preceding and following be welded into one, the welding action to be arrested by a period. For example, rn? n 2 / was to mean (w 2 n 2 )/(w 2 +w 2 ), and a/bed to mean , ,, but a/bc-d to mean ,- d. "This solidus notation for algebraic expressions oc- oc 1 Henri Cambuston, Definition de las principals opcraciones de arismetica (1843), p. 26. 2 Antonio Serra Y Oliveres, Manuel de la Tipografia Espanola (Madrid, 1852), p. 71. 3 G. G. Stokes, Math, and Phys. Papers, Vol. I (Cambridge, 1880), p, vii. See also J. Larmor, Memoirs and Scient. Corr. ofG. G. Stokes, Vol. I (1907), p. 397. 4 O. Stolz and J. A. Gmeiner, Theoretische Arithmetik (2d cd.; Leipzig, 1911), p. 81. 6 J. W. L. Glaisher, Messenger of Mathematics, Vol. II (1873), p. 109. 6 W. J. G. Karsten, Lehrbegrif der gesamten Mathematik, Vol. I (Greifswald, 1767), p. 50, 51, 55. 7 G. Peano, Lezioni di analisi infinitesimale, Vol. I (Torino, 1893), p. 2. 8 Alexander Macfarlane, Lectures on Ten British Physicists (New York, 1919), p. 100, 101. 314 A HISTORY OF MATHEMATICAL NOTATIONS curring in the text has since been used in the Encyclopaedia Britannica, in Wiedemann's^nnafen and quite generally in mathematical litera- ture." It was recommended in 1915 by the Council of the London Mathematical Society to be used in the current text. "The use of small fractions in the rnidst of letterpress," says Bryan, 1 "is often open to the objection that such fractions are difficult to read, and, moreover, very often do not come out clearly in printing. It is especially difficult to distinguish % from \ For this reason it would be better to confine the use of these fractions to such common forms as J, ^, f , |, and to use the nptation 18/22 for other fractions." SIGNS OP DECIMAL FRACTIONS 276. Stevin's notation. The invention of decimal fractions is usually ascribed to the Belgian Simon Stevin, in his La Disme, pub- lished in 1585 ( 162). But at an earlier date several other writers came so close to this invention, and at a later date other writers ad- vanced the same ideas, more or less independently, that rival candi- dates for the honor of invention were bound to be advanced. The La Disme of Stevin marked a full grasp of the nature and importance of decimal fractions, but labored under the burden of a clumsy nota- tion. The work did not produce any immediate effect. It was trans- lated into English by R. Norton 2 in 1608, who slightly modified the notation by replacing the circles by round parentheses. The frac- tion .3759 is given by Norton in the form 3 (1) 7 (2) 5 (3) 9 (4) . 277. Among writers who adopted Stevin's decimal notation is Wilhelm von Kalcheim 3 who writes 693 @ for our 6.93. He applies it also to mark the decimal subdivisions of linear measure: "Die Zeichen sind diese: (o) ist ein ganzes oder eine ruthe: ist ein erstes / prime oder schuh: @ ist ein zweites / secunde oder Zoll: ein drittes / korn oder gran: @ ist ein viertes stipflin oder minuten: und so forthan." Before this J. H. Beyer writes 4 8 798 for 8.00798; also 1 G. H. Bryan, Mathematical Gazette, Vol. VIII (1917), p. 220. 2 Disme: the Art of Tenths, or Decimall Arithmetike, .... invented by the excel- lent mathematician, Simon Stevin. Published in English with some additions by Robert Norton, Gent. (London, 1608). See also A. de Morgan in Companion to the British Almanac (1851), p. 11. 3 Zusammenfassung etlicher geomelrischen Aufgaben. .... Durch Wilhelra von Kalcheim, genant Lohausen Obristen (Bremen, 1629), p. 117. 4 Johann Hartmann Beyer, Logistica decimalis, das ist die Kunstrechnung mil den zehntheiligen Briichen (Frankfurt a/M., 1603). We have not seen Beyer's DECIMAL FRACTIONS 315 viii i ii iii iv v vi i ii iil iv v vi 14.3761 for 14.00003761, 123.4.5.9.8.7.2. or 123.4.5.9.8.7.2 or 123. 459. 872 for 123.459872, 643 for 0.0643. That Stevin's notation was not readily abandoned for a simpler one is evident from Ozanam's use 1 of a slight modification of it as (1) (2) (3) (4) (0) fl) (2) late as 1691, in passages like " T Wftr 6g. d 6 Q Q 7," and 3 9 8 for our 3.98. 278. Other notations used before 1617. Early notations which one might be tempted to look upon as decimal notations appear in works whose authors had no real comprehension of decimal fractions and their importance. Thus Regiomontanus, 2 in dividing 85869387 by 60000, marks off the last four digits in the dividend and then divides by 6 as follows: 8586|9387 1431 In the same way, Pietro Borgi 3 in 1484 uses the stroke in dividing 123456 by 300, thus "per 300 1 2 3 4 | 5 6 4 1 1 411JU-" Francesco Pellos (Pellizzati) in 1492, in an arithmetic published at Turin, used a point and came near the invention of decimal fractions. 4 Christoff Rudolff 5 in his Coss of 1525 divides 652 by 10. His words are: "Zu exempel / ich teile 652 durch 10. stet also 65/2. ist 65 der quocient vnnd 2 das iibrig. Kompt aber ein Zal durch 100 zu teilen / schneid ab die ersten zwo figuren / durch 1000 die erstcn drey / also weiter fur yede o ein figur." ("For example, I divide 652 by 10. It gives 65/2; thus, 65 is the quotient and 2 the remainder. If a number is to be divided by 100, cut off the first two figures, if by book ; our information is drawn from J. Tropfke, Geschichte der Elementar-Mathc- matik, Vol. I (2d ed.; Berlin and Leipzig, 1921), p. 143 ;S. Giinther, Geschichtc der Mathcmatik, Vol. I (Leipzig, 1908), p. 342. 1 J. Ozanam, V 'Usage du Compas de Proportion (a La Haye, 1691), p. 203, 211. 2 AbhantUungen zur Geschichtc der Mathematik, Vol. XII (1902), p. 202, 225. 3 See G. Enestrom in Bibliotheca mathematica (3d ser.), Vol. X (1909-10), p. 240. 4 D. E. Smith, Rara arithmetica (1898), p. 50, 52. 6 Quoted by J. Tropfke, op. cit., Vol. I (2d ed., 1921), p. 140. 316 A HISTORY OF MATHEMATICAL NOTATIONS 1,000 the first three, and so on for each a figure.") This rule for division by 10,000, etc., is given also by P. Apian 1 in 1527. In the Exempel Biichlin (Vienna, 1530), Rudolff performs a multiplication involving what we now would interpret as being deci- mal fractions. 2 Rudolff computes the values 375 (l+iiU) n for n=l, 2, ,10. For n = l he writes 393 | 75, which really denotes 393.75; for ft = 3 he writes 434 | 109375. The computation for n = 4 is as fol- lows: 434|109375 21 70546875 455|81484375 Here Rudolff uses the vertical stroke as we use the comma and, in passing, uses decimals without appreciating the importance and' generality of his procedure. F. Vieta fully comprehends decimal fractions and speaks of the advantages which they afford; 3 he approaches close to the modern notations, for, after having used (p. 15) for the fractional part smaller type than for the integral part, he separated the decimal from the integral part by a vertical stroke (p. 64, 65); from the vertical stroke to the actual comma there is no great change. In 1592 Thomas Masterson made a close approach to decimal frac- tions by using a vertical bar as separatrix when dividing 337652643 by a million and reducing the result to shillings and pence. He wrote : 4 facit /. 337 s. 1 3 d. 6 5 2 6 4 3 " 052860 634320 John Kepler in his Oesterreichisches Wein-Visier-Buchlein (Lintz, MDCXVI), reprinted in Kepler's Opera omnia (ed. Ch. Frisch), Volume V (1864), page 547, says: "Furs ander, weil ich kurtze Zahlen brauche, derohalben es offt Briiche geben wirdt, so mercke, dass alle Ziffer, welche nach dem Zeichen (() folgen, die gehoren zu 1 P. Apian, Kauffmannsz Rechnung (Ingolstadt, 1527), fol. cttjr . Taken from J. Tropfke, op. tit., Vol. I (2d ed., 1921), p. 141. 2 See D. E. Smith, "Invention of the Decimal Fraction," Teachers College Bulletin (New York, 1910-11), p. 18; G. Enestrom, Bibliotheca mathematica (3d ser.), Vol. X (1909-10), p. 243. 3 F. Vieta, Universalium inspcclionum, p. 7; Appendix to the Canon mathe- maticus (1st ed.; Paris, 1579). We copy this reference from the Encyclopedic des scienc. m.ath., Tome I, Vol. I (1904), p. 53, n. 180. 4 A. de Morgan, Companion to the British Almanac (1851), p. 8. DECIMAL FRACTIONS 317 dem Bruch, als der Zehlcr, der Nenner darzu wird nicht gesetzt, ist aber allczcit cine runde Zehnerzahl von so vil Nullen, als vil Ziffcr nach dem Zeichen kommen. Wann kcin Zeichen nicht ist, das ist eine gantze Zahl ohne Bruch, vnd wann also alle Ziffern nach dem Zeichen gehen, da hebcn sie bissweilen an von einer Nullen. Disc Art der Bruch-rechnung ist von Jost Biirgcn zu der sinusrechnung erdacht, vnd ist darzu gut, dass ich den Bruch abkiirtzen kan, wa er vnnotig lang werden wil, ohne sonderen Schaden der vberigen Zahlen; kan ihne auch etwa auff Erhaischung der Notdurfft crlengern. Item lesset sich also die gantze Zahl vnd der Bruch mit einander durch alle species Arithrneticae handlcn wie nur eine Zahl. Als wann ich rechne 365 Gulden mit 6 per cento, wievil bringt es dess Jars Inter- esse? dass stehet nun also : 3(65 6 mal facit21(90 vnd bringt 21 Gulden vnd 90 hundertheil, oder 9 zchentheil, das ist 54 kr." Joost Burgi 1 wrote 1414 for 141.4 and 001414 for 0.01414; on the o title-page of his Progress-Tabulen (Prag, 1620) he wrote 230270022 for our 230270.022. This small circle is referred to often in his Grundlicher Unterrichty first published in 1856. 2 279. Did Pitiscus use the decimal point? If Bartholomaeus Pitiscus of Heidelberg made use of the decimal point, he was probably the first to do so. Recent writers 3 on the history of mathematics are 1 See R. Wolf, Vicrtdj. Nalurf. Gcs. (Ziirich), Vol. XXXIII (1888), p. 226. 2 Gruncrt's Archiv der Mathcmatik und Physik, Vol. XXVI (1856), p. 316-34. 3 A. von Braunmuhl, Geschichte der Trigonometric, Vol. I (Leipzig, 1900), p. 225. M. Cantor, Vorlcsungen iiber Geschichte der Mathematik, Vol. II (2d ed.; Leipzig, 1913), p. 604, 619. G. Enestrom in Bibliotheca mathematica (3d ser.), Vol. VI (Leipzig, 1905), p. 108, 109. J. W. L. Glaisher in Napier Tercentenary Memorial Volume (London, 1913), p. 77. N. L. W. A. Gravclaar in Nieuw Archiefvoor Wiskunde (2d ser.; Amsterdam), Vol. IV (1900), p. 73. S. Giinther, Geschichte der Mathematik, 1. Toil (Leipzig, 1908), p. 342. L. C. Karpinski in Science (2d ser.), Vol. XLV (New York, 1917), p. 663-65. D. E. Smith in Teachers College Bulletin, Department of Mathematics (New York, 1910-11), p. 19. J. Tropfke, Geschichte der Elementar-Mathematik, Vol. I (2d ed. ; Leipzig, 1921), p. 143. 318 A HISTORY OF MATHEMATICAL NOTATIONS divided on the question as to whether or not Pitiscus used the decimal point, the majority of them stating that he did use it. This disagree- ment arises from the fact that some writers, apparently not having access to the 1608 or 1612 edition of the Trigonometric, 1 of Pitiscus, reason from insufficient data drawn from indirect sources, while others fail to carry conviction by stating their conclusions without citing the underlying data. Two queries are involved in this discussion: (1) Did Pitiscus employ decimal fractions in his writings? (2) If he did employ them, did he use the dot as the separatrix between units and tenths? Did Pitiscus employ decimal fractions? As we have seen, the need of considering -this question arises from the fact that some early writers used a symbol of separation which we could interpret as separating units from tenths, but which they themselves did not so interpret. For instance, 2 Christoff Rudolff in his Coss of 1525 divides 652 by 10, "stet also 65|2. ist 65 der quocientvnnd 2dasiibrig." The figure 2 looks like two-tenths, but in RudohTs mind it is only a re- mainder. With him the vertical bar served to separate the 65 from this remainder; it was not a decimal separatrix, and he did not have the full concept of decimal fractions. Pitiscus, on the other hand, did have this concept, as we proceed to show. In computing the chord of an arc of 30 (the circle having 10 7 for its radius), Pitiscus makes the statement (p. 44): "All these chords are less than the radius and as it were certain parts of the radius, which parts are com- monly written iVsWoVtr- But much more brief and necessary for the work, is this writing of it .05176381. For those numbers are alto- gether of the same value, as these two numbers 09. and -fa are." In the original Latin the last part reads as follows: " . . . . quae partes vulgo sic scriberentur iVoVoWo Sed multo compendiosior et ad calculum accommodatior est ista scriptio .05176381. Omnino autem idem isti numeri valent, sicut hi duo numeri 09. et VV idem valent." One has here two decimals. The first is written .05176381. The dot on the left is not separating units from tenths ; it is only a rhetorical mark. The second decimal fraction he writes 09., and he omits the dot on the left. The zero plays here the role of decimal separatrix. 1 1 have used the edition of 1612 which bears the following title: Bartholamci \ Pitisci Grunbergensis \ Silesij \ Trigonometriae \ Sioe. De dimensione Triangulor [urn] Libri Qvinqve. Jtem \ Problematvm variorv. [m] nempe \ Geodaeticorum, \ Alti- metricorum, \ Geographicorum, \ Gnom&nicorum, et \ Astronomicorum: \ Libri Decem. \ Editio Tertia. \ Cui recens accessit Pro \ blematum Arckhiteclonicarum Liber \ unus \ Franeofurti. \ Typis Nicolai Hofmanni: \ Surnptibus lonae Rosae] M.DCXIL 2 Quoted from J. Tropfke, op. tit., Vol. I (1921), p. 140. DECIMAL FRACTIONS 319 The dots appearing here are simply the punctuation marks written after (sometimes also before) a number which appears in the running text of most medieval manuscripts and many early printed books on mathematics. For example, Clavius 1 wrote in 1606: "Deinde quia minor est \. quam $. erit per propos .8. minutarium libri 9. Euclid, minor proportio 4. ad 7. quam 3. ad 5." Pitiscus makes extensive use of decimal fractions. In the first five books of his Trigonometria the decimal fractions are not preceded by integral values. The fractional numerals are preceded by a zero; thus on page 44 he writes 02679492 (our 0.2679492) and finds its square root which he writes 05176381 (our 0.5176381). Given an arc and its chord, he finds (p. 54) the chord of one-third that arc. This leads to the equation (in modern symbols) 3o?x 3 = . 5176381, the radius being unity. In the solution of this equation by approximation he obtains successively 01, 017, 0174 .... and finally 01743114. In computing, he squares and cubes each of these numbers. Of 017, the square is given as 00289, the cube as 0004913. This proves that Pitiscus understood operations with decimals. In squaring 017 ap- pears the following: "001.7 2 7 1 89 002 89.4" What role do these dots play? If we put a = - 1 V> & vJk> then (a+6)2 = 2+(2a+6)6; 001 -a 2 , 027=(2a+6), 00189= (2a+&)6, 00289= (a+fr). 2 The dot in 001.7 serves simply as a separator be- tween the 001 and the digit 7, found in the second step of the approxi- mation. Similarly, in 00289.4, the dot separates 00289 and the digit 4, found in the third step of the approximation. It is clear that the dots used by Pitiscus in the foreging approximation are not decimal points. The part of Pitiscus' Trigonometria (1612) which bears the title "Problematvm variorvm .... libri vndecim" begins a new pagina- tion. Decimal fractions are used extensively, but integral parts appear and a vertical bar is used as decimal separatrix, as (p. 12) where he says, "pro .... 13|00024. assumo 13. fractione scilicet 10 oo oa neglecta." ("For 13.00024 I assume 13, the fraction, namely, fbo.Voo being neglected/') Here again he displays his understanding of decimals, and he uses the dot for other purposes than a decimal separatrix. The writer has carefully examined every appearance of 1 Christophori Clavius .... Geometria practica (Mogvntiac, 160G), p. 343- 320 A HISTORY OF MATHEMATICAL NOTATIONS dots in the processes of arithmetical calculation, but has failed to find the dot used as a decimal separatrix. There are in the Pitiscus of 1612 three notations for decimal fractions, the three exhibited in 0522 (our .522), 5 1 269 (our 5.269), and the form (p. 9) of common fractions, 121-jVoV In one case (p. 11) there occurs the tautological notation 29|-,Vjr (our 29.95). 280. But it has been affirmed that Pitiscus used the decimal point in his trigonometric Table. Indeed, the dot does appear in the Table of 1612 hundreds of times. Is it used as a decimal point? Let us quote from Pitiscus (p. 34) : "Therefore the radius for the making of these Tables is to be taken so much the more, that there may be no error in so many of the figures towards the left hand, as you will have placed in the Tables: And as for the superfluous numbers they are to be cut off from the right hand toward the left, after the ending of the calculation. So did Regiomontanus, when he would calculate the tables of sines to the radius of 6000000; he took the radius 60000000000. and after the computation was ended, he cut off from every sine so found, from the right hand toward the left four figures, so Rhaeticus when he would calculate a table of sines to the radius of 10000000000 took for the radius 1000000000000000 and after the calculation was done, he cut off from every sine found from the right hand toward the left five figures: But I, to find out the numbers in the beginning of the Table, took the radius of 100000 00000 00000 00000 00000. But in the Canon itself have taken the radius divers numbers for necessity sake: As hereafter in his place shall be declared." On page 83 Pitiscus states that the radius assumed is unity fol- lowed by 5, 7, 8, 9, 10, 11, or 12 ciphers, according to need. In solving problems he takes, on page 134, the radius 10 7 and writes sin 6146' = 8810284 (the number in the table is 88102.838); on page 7 ("Probl. var.") he takes the radius 10 5 and writes sin 4110' = 65825 (the num- ber in the Table is 66825.16). Many examples are worked, but in no operation are the trigonometric values taken from the Table written down as decimal fractions. In further illustration we copy the fol- lowing numerical values from the Table of 1612 (which contains sines, tangents, and secants) : " sin 2" = 97 sec 3" = 100000 . 00001 . 06 sin 3" = 1 . 45 sec 230' = 100095 . 2685 . tan 3" = 1 . 45 sec 330' = 100186 . 869 sin 8959'59" = 99999 . 99999 . 88 tan 8959'59" = 20626480624 . sin 3031' = 50778 . 90 sec 3031' = 116079 . 10*' DECIMAL FRACTIONS 321 To explain all these numbers the radius must be taken 10 12 . The 100000.00001.06 is an integer. The dot on the right is placed be- tween tens and hundreds. The dot on the left is placed between millions and tens of millions. When a number in the Table contains two dots, the left one is always between millions and tens of millions. The right-hand dot is be- tween tens and hundreds, except in the case of the secants of angles be- tween 019' and 231 / and in the case of sines of angles between 8759' and 8940'; in these cases the right-hand dot is placed (probably through a printer's error) between hundreds and thousands (see sec. 230'). The tangent of 8959'59" (given above) is really 20626480624- 0000000, when the radius is 10 12 . All the figures below ten millions are omitted from the Table in this and similar cases of large functional values. If a sine or tangent has one dot in the Table and the secant for the same angle has two dots, then the one dot for the sine or tangent lies between millions and tens of millions (see sin 3", sec 3"). If both the sine and secant of an angle have only one dot in the Table and r= 10 12 , that dot lies between millions and tens of millions (see sin 3031' and sec 3031'). If the sine or tangent of an angle has no dots whatever (like sin 2"), then the figures are located immedi- ately below the place for tens of millions. For all angles above 230' and below 88 the numbers in the Table contain each one and only one dot. If that dot were looked upon as a decimal point, correct re- sults could be secured by the use of that part of the Table. It would imply that the radius is always to be taken 10 5 . But this interpreta- tion is invalid for any one of the following reasons: (1) Pitiscus does not always take the r = 10 5 (in his early examples he takes r= 10 7 ), and he explicitly says that the radius may be taken 10 5 , 10 7 , 10 8 , 10 9 , 10 10 , 10 11 , or 10 12 , to suit the degrees of accuracy demanded in the solution. (2) In the numerous illustrative solutions of problems the numbers taken from the Table are always in integral form. (3) The two dots appearing in some numbers in the Table could not both be decimal points. (4) The numbers in the Table containing no dots could not be integers. The dots were inserted to facilitate the selection of the trigono- metric values for any given radius. For r= 10 5 , only the figures lying to the left of the dot between millions and tens of millions were copied. For r=10 l , the figures to the left of the dot between tens and hun- dreds were chosen, zeroes being supplied in cases like sin 3031', where there was only one dot, so as to yield sin 3031' = 5077890000. 322 A HISTORY OF MATHEMATICAL NOTATIONS For r = 10 7 , the figures for 10 5 and the two following figures were copied from the Table, yielding, for example, sin 3031' = 5077890. Similarly for other cases. In a Table 1 which Pitiscus brought out in 1613 one finds the sine of 252 / 30" given as 5015.71617.47294, thus indicating a different place assignment of the dots from that of 1612. In our modern tables the natural sine of 252 / 30" is given as .05015. This is in harmony with the statement of Pitiscus on the title-page that the Tables are computed "ad radium 1.00000.00000.00000." The observation to be stressed is that these numbers in the Table of Pitiscus (1613) are not decimal fractions, but integers. Our conclusions, therefore, are that Pitiscus made extended use of decimal fractions, but that the honor of introducing the dot as the separatrix between units and tenths must be assigned to others. J. Ginsburg has made a discovery of the occurrence of the dot in the position of a decimal separatrix, which he courteously permits to be noted here previous to the publication of his own account of it. He has found the dot in Clavius' Astrolabe, published in Rome in 1593, where it occurs in a table of sines and in the explanation of that table (p. 228). The table gives sin 1612' = 2789911 and sin 1613' = 2792704. Clavius places in a separate column 46.5 as a cor- rection to be made for every second of arc between 1612' and 1613'. He obtained this 46.5 by finding the difference 2793 "between the two sines 2789911.2792704," and dividing that difference by 60. He identifies 46.5 as signifying 46 t *V This dot separates units and tenths. In his works, Clavius uses the dot regularly to separate any two suc- cessive numbers. The very sentence which contains 46.5 contains also the integers "2789911.2792704." The question arises, did Clavius in that sentence use both dots as general separators of two pairs of numbers, of which one pair happened to be the integers 46 and the five-tenths, or did Clavius consciously use the dot in 46.5 in a more restricted sense as a decimal separatrix? His use of the plural "duo hi numeri 46,5" goes rather against the latter interpretation. If a more general and more complete statement can be found in Clavius, these doubts may be removed. In his Algebra of 1608, Clavius writes all decimal fractions in the form of common fractions. Nevertheless, Clavius unquestionably deserves a place in the history of the intro- duction of the dot as a decimal separatrix. More explicit in statement was John Napier who, in his Rabdologia 1 B. Pitiscus, Thesavrvs mathematicvs, sive Canon sinwn (Francofurti, 1613), p. 19. DECIMAL FRACTIONS 323 of 1617, recommended the use of a "period or comma" and uses the comma in his division. Napier's Construdio (first printed in 1619) was written before 1617 (the year of his death). In section 5 he says: " Whatever is written after the period is a fraction," and he actually uses the period. In the Leyden edition of the Construdio (1620) one finds (p. 6) "25.803. idem quod 25M\." 281. The point occurs in E. Wright's 1616 edition of Napier's Description but no evidence has been advanced, thus far, to show that the sign was intended as a separator of units and tenths, and not as a more general separator as in Pitiscus. 282. The decimal comma and point of Napier. That John Napier in his Rabdologia of 1617 introduced the comma and point as sepa- rators of units and tenths, and demonstrated that the comma was intended to be used in this manner by performing a division, and properly placing the comma in the quotient, is admitted by all his- torians. But there are still historians inclined to the belief that he was not the first to use the point or comma as a separatrix between units and tenths. We copy from Napier the following: "Since there is the same facility in working with these fractions as with whole numbers, you will be able after completing the ordinary division, and adding a period or comma, as in the margin, to add to the dividend or to the remainder one cypher to obtain 6 4 1 3 6 3 1 6 1 1 8,0 1 4 1 402 429 8 6 1 9 4,0 0(1 9 9 3,2 7 3 432 3888 3888 1296 864 3024 .1296 tenths, two for hundredths, three for thousandths, or more after- wards as required: And with these you will be able to proceed with the working as above. For instance, in the preceding example, here repeated, to which we have added three cyphers, the quotient will 324 A HISTORY OF MATHEMATICAL NOTATIONS become 1993,273, which signifies 1993 units and 273 thou- sandth parts or WoV" 1 Napier gives in the Rabdologia only three examples in which decimals occur, and even here he uses in the text the sexagesimal ex- ponents for the decimals in the statement of the results. 2 Thus he / // /// //// writes 1994.9160 as 1994,9 1 6 ; in the edition brought out at Leyden in 1626, the circles used by S. Stevin in his notation of deci- mals are used in place of Napier's sexagesimal exponents. Before 1617, Napier used the decimal point in his Constructio, where he explains the notation in sections 4, 5, and 47, but the Con- structio was not published until 1619, as already stated above. In section 5 he says: "Whatever is written after the period is a fraction," and he actually uses the period. But in the passage we quoted from Rabdologia he speaks of a "period or comma" and actually uses a com- ma in his illustration. Thus, Napier vacillated between the period and the comma; mathematicians have been vacillating in this matter ever since. In the 1620 edition 3 of the Constructio, brought out in Leyden, one reads: "Vt 10000000.04, valet idem, quod 1 0000000 r U- Item 25.803. idem quod 25 ^V Item 9999998.0005021, idem valet quod 9999998 iT H-Uinr- & sic de caeteris." 283. Seventeenth-century notations after 1617. The dot or comma attained no ascendancy over other notations during the seventeenth century. In 1623 John Johnson (the survaighour)* published an Arithmatick which stresses decimal fractions and modifies the notation of Stevin by omitting the circles. Thus, 3. 2 2 9 1 6 is written 1. 2. 3. 4. 5. 3 22916, while later in the text there occurs the symbolism 31 | 2500 and 54)2625, and also the more cautious "358149411 fifths" for our 358.49411. 1 John Napier, Rabdologia (Edinburgh, 1617), Book I, chap. iv. This passage is copied by W. R. Macdonald, in his translation of John Napier's Constructio (Edinburgh, 1889), p. 89. 2 J. W. L. Glashier, "Logarithms and Computation," Napier Tercentenary Memorial Volume (ed. Cargill Gilston Knott; London, 1915), p. 78. 3 Mirifwi logarithmorvm Canonis Constructio .... authore & Inventore loanne Nepero, Barone Merchistonii, etc, (Scoto. Lvgdvni, M.DC.XX.), p. 6. 4 From A. de Morgan in Companion to the British Almanac (1851), p. 12. DECIMAL FRACTIONS 325 Henry Briggs 1 drew a horizontal line under the numerals in the decimal part which appeared in smaller type and in an elevated posi- tion; Briggs wrote 5 9J ^ for our 5.9321. But in his Tables of 1624 he employs commas, not exclusively as a decimal separatrix, although one of the commas used for separation falls in the right place between units and tenths. He gives -0,22724,3780 as the logarithm of ff. A. Girard 2 in his Invention nouvelle of 1629 uses the comma on one occasion ; he finds one root of a cubic equation to be IfVW and then explains that the three roots expressed in decimals are 1,532 and 347 and 1,879. The 347 is .347; did Girard consider the comma un- necessary when there was no integral part? Burgi's and Kepler's notation is found again in a work which appeared in Poland from the pen of Joach. Stegman; 3 he writes 39(063. It occurs again in a geometry written by the Swiss Joh. Ardiiser. 4 William Oughtrcd adopted the sign 2|5 in his Clavis mathematicae of 1631 and in his later publications. In the second edition of Wingate's Arithmetic (1650; ed. John Kersey) the decimal point is used, thus: .25, .0025. In 1651 Robert Jager 5 says that the common way of natural arith- metic being tedious and prolix, God in his mercy directed him to what he published; he writes upon decimals, in which 16|7249 is our 16.7249. Richard Balam 6 used the colon and wrote 3:04 for our 3.04. This same symbolism was employed by Richard Rawly ns, 7 of Great Yar- mouth, in England, and by H. Meissner 8 in Germany. 1 Henry Briggs, Arithmetica logarithmica (London, 1624), Lectori. S. 2 De Morgan, Companion to the British Almanac (1851), p. 12; Invention nou- velle, fol. E2. 3 Joach. Stegman, Institutionum mathematicarum libri II (Rakow, 1630), Vol. I, cap. xxiv, "De logistica decimali." We take this reference from J. Tropfke, op. cit., Vol. I (2d ed., 1921), p. 144. 4 Joh. Ardttser, Geometriae theoricae et practicae XII libri (Zurich, 1646), fol. 306, 1SO&, 27()a. 5 Robert Jager, Artificial Arithmclick in Decimals (London, 1651). Our infor- mation is drawn from A. de Morgan in Companion to the British Almanac (1851), p. 13. 6 Rich. Balam, Algebra (London, 1653), p. 4. 7 Richard Rawlyns, Practical Arilhmetick (London, 1656), p. 262. 8 H. Meissner, Geometria tyronica (1696[?]). This reference is taken from J. Tropfke, op. cit.. Vol. I (2d ed!, 1921), p. 144. 326 A HISTORY OF MATHEMATICAL NOTATIONS Sometimes one encounters a superposition of one notation upon another, as if one notation alone might not be understood. Thus F. van Schooten 1 writes 58,5 for 58.5, and 638,82 for 638.82. Tobias 11 111 IV V Beutel 2 writes 645. JJfo. A. Tacquet 3 sometimes writes 25.8 0079, at other times omits the dot, or the Roman superscripts. Samuel Foster 4 of Gresham College, London, writes 31. k^; he does not rely upon the dot alone, but adds the horizontal line found in Briggs. Johann Caramuel 5 of Lobkowitz in Bohemia used two horizontal parallel lines, like our sign of equality, as 22 = 3 for 22.3, also 92 = 123,345 for 92.123345. In a Parisian text by Jean Prestet 6 272097792 is given for 272.097792; this mode of writing had been sometimes used by Stevin about a century before Prestet, and in 1603 by Beyer. William Molyneux 7 of Dublin had three notations; he frequently used the comma bent toward the right, as in 30 t 24. N. Mercator 8 in his Logarithmotechnia and Dechales 9 in his course of mathematics used the notation as in 12[345. 284. The great variety of forms for separatrix is commented on by Samuel Jeake in 1696 as follows: "For distinguishing of the Decimal Fraction from Integers, it may truly be said, Quot Homines, lot Sen- tentiae; every one fancying severally. For some call the Tenth Parts, the Primes; the Hundredth Parts, Seconds; the Thousandth Parts, Thirds, etc. and mark them with Indices equivalent over their heads. As to express 34 integers and iW<ftr Parts of an Unit, they do it thus, / // /// //// (1) (2) (3) (4) 34.1. 4. 2. 6. Or thus, 34.1. 4. 2. 6. Others thus, 34,1426""; or thus, 34,1426^ 4 >. And some thus, 34.1 . 4 . 2 . 6 . setting the Decimal Parts 1 Francisci a Schooten, Exercitalionvm mathemaiicarum liber primus (Leyden, 1657), p. 33, 48, 49. 2 Tobias Beutel, Geometrischer Lust-Garten (Leipzig, 1690), p. 173. 3 Arithmeticae theoria et praxis, autore Andrea Tacqvet (2d ed.; Antwerp, 1665), p. 181-88. 4 Samuel Foster, Miscellanies: or Mathematical Lvcvbrations (London, 1659), p. 13. 6 Joannis Caramvels Mathesis Biceps. Vetus, et Nova (Companiae, 1670), "Arithmetica," p. 191. 6 Jean Prestet, Nouveaux elcmens des mathematiques, Premier volume (Paris, 1689), p. 293. 7 William Molyneux, Treatise of Dioplricks (London, 1692), p. 165. 8 N. Mercator, Logarithmotechnia (1668), p. 19. 9 A. de Morgan, Companion to the British Almanac (1851), p. 13. DECIMAL FRACTIONS 327 at little more than ordinary distance one from the other Others distinguish the Integers from the Decimal Parts only by placing a Coma before the Decimal Parts thus, 34,1426; a good way, and very useful. Others draw a Line under the Decimals thus, 34 L4 - 3 - 6 -, writing them in smaller Figures than the Integers. And others, though they use the Coma in the work for the best way of distinguishing them, yet after the work is done, they use a Rectangular Line after the place of the Units, called Separatrix, a separating Line, because it separates the Decimal Parts from the Integers, thus 34 [1426. And sometimes the Coma is inverted thus, 34' 1426, contrary to the true Coma, and set at top. I sometimes use the one, and sometimes the other, as cometh to hand." The author generally uses the comma. This detailed state- ment from this seventeenth-century writer is remarkable for the omission of the point as a decimal separatrix. 285. Eighteenth-century discard of clumsy notations. The chaos in notations for decimal fractions gradually gave way to a semblance of order. The situation reduced itself to trials of strength between the comma and the dot as separatrices. To be sure, one finds that over a century after the introduction of the decimal point there were authors who used besides the dot or comma the strokes or Roman numerals to indicate primes, seconds, thirds, etc. Thus, Chelucci 1 in 1738 writes o i n in iv i iv ii v 5.8 6 4 2, also4.2 5 for 4.2005, 3.5 7for3.05007. W. Whiston 2 of Cambridge used the semicolon a few times, as in 0;9985, though ordinarily he preferred the comma. O. Gherli 8 in Modena, Italy, states that some use the sign 35 1 345, but he himself uses the point. E. Wells 4 in 1713 begins with 75.25, but later in his arithmetic introduces Oughtred's J75. Joseph Raphson's transla- tion into English of I. Newton's Universal Arithmetick (1728) , 5 con- tains 732,[569 for our 732.569. L'Abb6 Deidier 6 of Paris writes the 1 Paolirio Chelucci, Institutions analyticae .... auctore Paulino A. S. Josepho Lucensi (Rome), p. 35, 37, 41, 283. 2 Isaac Newton, Arithmetica Vniversalis (Cambridge, 1707), edited by G. W[histon], p. 34. 3 O. Gherli, Gli dementi .... delle mathematiche pure, Vol. I (Modena, 1770), p. 60. 4 Edward Wells, Young gentleman's arithmetick (London, 1713), p. 59, 105, 157. 6 Universal Arithmetick, or Treatise of Arithmetical Composition and Resolu- tion .... transl. by the late Mr. Joseph Ralphson, & revised and corrected by Mr. Cunn (2d ed.; London, 1728), p. 2. 6 L'Abbe" Deidier, U Arithmetique des Gtometres, ou nouveaux eUmens de mathe'- matiques (Paris, 1739), p. 413. 328 A HISTORY OF MATHEMATICAL NOTATIONS decimal point and also the strokes for tenths, hundredths, etc. He says: "Pour ajouter ensemble 32.6' 3" 4'" et 8.5' 4".3'" 32 6 3 4 111 854 3 111 41 1 7 7 m " A somewhat unusual procedure is found in Sherwin's Tables 1 of 1741, where a number placed inside a parenthesis is used to designate the number of zeroes that precede the first significant figure in a decimal ; thus, (4) 2677 means .00002677. In the eighteenth century, trials of strength between the comma and the dot as the separatrix were complicated by the fact that Leib- niz had proposed the dot as the symbol of multiplication, a proposal which was championed by the German textbook writer Christian Wolf and which met with favorable reception throughout the Conti- nent. And yet Wolf 2 himself in 1713 used the dot also as separatrix, as "loco 5i ffVinF scribimus 5.0047." As a symbol for multiplication the dot was seldom used in England during the eighteenth century, Oughtred's X being generally preferred. For this reason, the dot as a separatrix enjoyed an advantage in England during the eighteenth century which it did not enjoy on the Continent. Of fifteen British books of that period, which we chose at random, nine used the dot and six the comma. In the nineteenth century hardly any British authors employed the comma as separatrix. In Germany, France, and Spain the comma, during the eighteenth century, had the lead over the dot, as a separatrix. During that century the most determined continental stand in favor of the dot was made in Belgium 3 and Italy. 4 But in recent years the comma has finally won out in both countries. 1 H. Sherwin, Mathematical Tables (3d ed. ; rev. William Gardiner, London, 1741), p. 48. 2 Christian Wolf, Elementa matheseos universae, Tomus I (Halle, 1713), p. 77. 3 De'sire' Andre 4 , Des Notations MatMmatiques (Paris, 1909), p. 19, 20. 4 Among eighteenth-century writers in Italy using the dot are Paulino A. S. Josepho Lucensi who in his Institutiones analyticae (Rome, 1738) uses it in con- nection with an older symbolism, "3.05007"; G. M. della Torre, Istituzioni arim- metiche (Padua, 1768) ; Odoardo Gherli, Elementi delle matematiche pure, Modena, Tomo I (1770); Peter Ferroni, Magnitudinum exponentialium logarithmorum et trigonometriae sublimis theoria (Florence, 1782); F. A. Tortorella, Arithmetica degVidioti (Naples, 1794). DECIMAL FRACTIONS 329 286. Nineteenth century: different positions for dot and comma. In the nineteenth century the dot became, in England, the favorite separatrix symbol. When the brilliant but erratic Randolph Churchill critically spoke of the "damned little dots," he paid scant respect to what was dear to British mathematicians. In that century the dot came to serve in England in a double capacity, as the decimal symbol and as a symbol for multiplication. Nor did these two dots introduce confusion, because (if we may use a situation suggested by Shakespeare) the symbols were placed in Romeo and Juliet positions, the Juliet dot stood on high, above Romeo's reach, her joy reduced to a decimal over his departure, while Romeo below had his griefs multiplied and was "a thousand times the worse" for want of her light. Thus, 25 means 2^, while 2.5 equals 10. It is difficult to bring about a general agreement of this kind, but it was achieved in Great Britain in the course of a little over half a century. Charles Hutton 1 said in 1795: "I place the point near the upper part of the figures, as was done also by Newton, a method which prevents the separatrix from being confounded with mere marks of punctuation." In the Latin edition 2 of Newton's Arithmetica uni- versalis (1707) one finds, "Sic numerus 732'|569. denotat septingentas triginta duas imitates, .... qui et sic 732,|569, vel sic 732*569. vel ctiam sic 732j569, nunnunquam scribitur .... 57104*2083 .... 0'064." The use of the comma prevails; it is usually placed high, but not always. In Horsely's and Castillon's editions of Newton's Arith- metica universalis (1799) one finds in a few places the decimal nota- tion 3572; it is here not the point but the comma that is placed on high. Probably as early as the time of Hutton the expression "deci- mal point" had come to be the synonym for "separatrix" and was used even when the symbol was not a point. In most places in Hors-^ ley's and Castillon's editions of Newton's works, the comma 2,5 is used, and only in rare instances the point 2.5. The sign 2 ' 5 was used in England by H. Clarke 3 as early as 1777, and by William Dickson 4 in 1800. After the time of Hutton the 2 5 symbolism was adopted by Peter Barlow (1814) and James Mitchell (1823) in their mathematical dictionaries. Augustus de Morgan states in his Arithmetic: "The 1 Ch. Hutton, Mathematical and Philosophical Dictionary (London, 1795), art. "Decimal Fractions." 2 1. Newton, Arithmetica universalis (ed. W. Whiston; Cambridge, 1707), p. 2. Sec also p. 15, 16. 3 H. Clarke, Rationale of Circulating Numbers (London, 1777). 4 W. Dickson in Philosophical Transactions, Vol. VIII (London, 1800), p. 231. 330 A HISTORY OF MATHEMATICAL NOTATIONS student is recommended always to write the decimal point in a line with the top of the figures, or in the middle, as is done here, and never at the bottom. The reason is that it is usual in the higher branches of mathematics to use a point placed between two numbers or letters which are multiplied together." 1 A similar statement is made in 1852 by T. P. Kirkman. 2 Finally, the use of this notation in Todhunter's texts secured its general adoption in Great Britain. The extension of the usefulness of the comma or point by assign- ing it different vertical positions was made in the arithmetic of Sir Jonas Moore 3 who used an elevated and inverted comma, 116*64. This notation never became popular, yet has maintained itself to the present time. Daniel Adams, 4 in New Hampshire, used it, also Juan de Dios Salazar 5 in Peru, Don Gabriel Ciscar 6 of Mexico, A. de la Rosa Toro 7 of Lima in Peru, and Federico Villareal 8 of Lima. The elevated and inverted comma occurs in many, but not all, the articles using decimal fractions in the Enciclopedia-vniversal ilvstrada Evropeo-Americana (Barcelona, 1924). Somewhat wider distribution was enjoyed by the elevated but not inverted comma, as in 2'5. Attention has already been called to the occurrence of this symbolism, a few times, in Horsley's edition of Newton's Arithmetica universalis. It appeared also in W. Winston's edition of the same work in 1707 (p. 15). Juan de Dios Salazar of Peru, who used the elevated inverted comma, also uses this. It is Spain and the Spanish-American countries which lead in the use of this notation. De La-Rosa Toro, who used the inverted comma, also used this. The 2'5 is found in Luis Monsante 9 of Lima; in Maximo 1 A. de Morgan, Elements of Arithmetic (4th cd.; London, 1840), p. 72. 2 T. P. Kirkman, First Mnemonical Lessons in Geometry, Algebra and Trigo- nometry (London, 1852), p. 5. 3 Moore's Arithmetick: In Four Books (3d ed.; London, 1688), p. 369, 370, 465. 4 Daniel Adams, Arithmetic (Keene, N.H., 1827), p. 132. 5 Juan de Dios Salazar, Lecciones de Aritmetica, Teniente del Cosmografo major de esta Republica del Peru (Arequipa, 1827), p. 5, 74, 126, 131. This book has three diiTerent notations: 2,5; 2'5; 2*5. 6 Don Gabriel Ciscar, Curso de esludios elementales de Marina (Mexico, 1825). 7 Agustin de La-Rosa Toro, Aritmetica Teorico-Proxtica (tercera ed. ; Lima, 1872), p. 157. 8 D. Federico Villareal, Calculo Binomial (P. I. Lima [Peru], 1898), p. 416. 9 Luis Monsante, Lecciones de Aritmetica Demostrada (7th ed. ; Lima, 1872), p. 89. DECIMAL FRACTIONS 331 Vazquez 1 of Lima; in Manuel Torres Torija 2 of Mexico; in D. J. Cortazar 3 of Madrid. And yet, the Spanish-speaking countries did not enjoy the monopoly of this symbolism. One finds the decimal comma placed in an elevated position, 2'5, by Louis Bertrand 4 of Geneva, Switzerland. Other writers use an inverted wedge-shaped comma, 5 in a lower position, thus: 2^5. In Scandinavia and Denmark the dot and the comma have had a very close race, the comma being now in the lead. The practice is also widely prevalent, in those countries, of printing the decimal part of a number in smaller type than the integral part. 6 Thus one frequently finds there the notations 2, 5 and 2. 5 . To sum up, in books printed within thirty-five years we have found the decimal notations 7 2-5, 2*5, 2,5, 2'5, 2*5, 2 A 5, 2, 5 , 2. 6 . 287. The earliest arithmetic printed on the American continent which described decimal fractions came from the pen of Greenwood, 8 professor at Harvard College. He gives as the mark of separation "a Comma, a Period, or the like," but actually uses a comma. The arith- metic of "George Fisher" (Mrs. Slack), brought out in England, and also her The American Instructor (Philadelphia, 1748) contain both the comma and the period. Dil worth's The Schoolmaster's Assistant, an English book republished in America (Philadelphia, 1733), used the period. In the United States the decimal point 9 has always had the 1 Maximo Vazquez, Aritmetica practica (septiema ed.; Lima, 1875), p. 57. 2 Manuel Torres Torija, Nociones de Algebra Superior y elemcntos fundamen- tales de cdlculo differencial e Integral (Mexico, 1894), p. 137. 3 D. J. Cortazdr, Tratado de Aritmetica (42d ed.; Madrid, 1904). 4 L. Bertrand, Developpment nouveaux de la partie elemenlaire des mathe- matiques, Vol. I (Geneva, 1778), p. 7. 5 As in A. F. Vallin, Aritmetica para los ninos (41st ed.; Madrid, 1889), p. 66. 6 Gustaf Haglund, Samlying of Ofningsexempel till Ldrabok i Algebra, Fjerde Upplagan (Stockholm, 1884), p. 19; Ofversigt af Kongl. Vetenskaps-Akademiens Forhandlingar, Vol. LIX (1902; Stockholm, 1902, 1903), p. 183, 329; Oversigl over del Kongelige Danske Videnskabernes Selskabs, Fordhandlinger (1915; Kobenhavn, 1915), p. 33, 35, 481, 493, 545. 7 An unusual use of the elevated comma is found in F. G. Gausz's Funfstellige vollstdndige Logar. u. Trig. Tafeln (Halle a. S., 1906), p. 125; a table of squares of numbers proceeds from AT = 0'00 to N = lO'OO. If the square of 03 is wanted, take the form 6'3; its square is 39'6900. Hence 63 2 = 3969. 8 Isaac Greenwood, Arithmctick Vulgar and Decimal (Boston, 1729), p. 49. See facsimile of a page showing decimal notation in L. C. Karpinski, History of Arithmetic (Chicago, New York, 1925), p. 134. 9 Of interest is Chauncey Lee's explanation in his American Accomptant (Lasingburgh, 1797), p. 54, that, in writing denominate numbers, he separates 332 A HISTORY OF MATHEMATICAL NOTATIONS lead over the comma, but during the latter part of the eighteenth and the first half of the nineteenth century the comma in the position of 2,5 was used quite extensively. During 1825-50 it was the influence of French texts which favored the comma. We have seen that Daniel Adams used 2'5 in 1827, but in 1807 he 1 had employed the ordinary 25,17 and ,375. Since about 1850 the dot has been used almost ex- clusively. Several times the English elevated dot was used in books printed in the United States. The notation 2*5 is found in Thomas Sarjeant's Arithmetic? in F. Nichols' Trigonometry? in American editions of Hutton's Course of Mathematics that appeared in the in- terval 1812-31, in Samuel Webber's Mathematics? in William Griev's Mechanics Calculator, from the fifth Glasgow edition (Philadelphia, 1842), in The Mathematical Diary of R. Adrain 5 about 1825, in Thomas Sherwin's Common School Algebra (Boston, 1867; 1st ed., 1845), in George R. Perkins' Practical Arithmetic (New York, 1852). Sherwin writes: "To distinguish the sign of Multiplication from the period used as a decimal point, the latter is elevated by inverting the type, while the former is larger and placed down even with the lower extremities of the figures or letters between which it stands." In 1881 George Bruce Halsted 6 placed the decimal point halfway up and the multiplication point low. It is difficult to assign definitely the reason why the notation 2*5 failed of general adoption in the United States. Perhaps it was due to mere chance. Men of influence, such as Benjamin Peirce, Elias Loomis, Charles Davies, and Edward Olncy, did not happen to be- come interested in this detail. America had no one of the influence of De Morgan and Todhunter in England, to force the issue in favor of 2*5. As a result, 2. 5 had for a while in America a double meaning, namely, 2 5/10 and 2 times 5. As long as the dot was seldom used to the denominations "in a vulgar table" by two commas, but "in a decimal table" by the decimal point; he writes 175,, 15,, 9, and 1.41. Daniel Adams, Scholar's Arithmetic (4th ed.; Keene, N.H., 1807). 2 Thomas Sarjeant, Elementary Principles of Arithmetic (Philadelphia, 1788), p. 80. 3 F. Nichols, Plane and Spherical Trigonometry (Philadelphia, 1811), p. 33. 4 Samuel Webber, Mathematics, Vol. I (Cambridge, 1801; also 1808, 2d cd.), p. 227. 6 R. Adrain, The Mathematical Diary, No. 5, p. 101. 6 George Bruce Halsted, Elementary Treatise on Mensuration (Boston, 1881). DECIMAL FRACTIONS 333 express multiplication, no great inconvenience resulted, but about 1880 the need of a distinction arose. The decimal notation was at that time thoroughly established in this country, as 2.5, and the dot for multiplication was elevated to a central position. Thus with us 2-5 means 2 times 5. Comparing our present practice with the British the situation is this: We write the decimal point low, they write it high; we place the multiplication dot halfway up, they place it low. Occasionally one finds the dot placed high to mark multiplication also in German books, as, for example, in Friedrich Meyer 1 who writes 2 ' 3 = 6. 288. It is a notable circumstance that at the present time the modern British decimal notation is also the notation in use in Austria where one finds the decimal point placed high, but the custom does not seem to prevail through any influence emanating from England. In the eighteenth century P. Mako 2 everywhere used the comma, as in 3,784. F. S. Mozhnik 3 in 1839 uses the comma for decimal fractions, as in 3,1344, and writes the product "2 . 3..n." The Sitzungsberichte der philosophisch-historischen Classe d. K. Akademie der Wissenschaften, Erster Band (Wien, 1848), contains decimal fractions in many articles and tables, but always with the low dot or low comma as decimal scparatrix ; the low dot is used also for multiplication, as in "1.2.3. . .r." But the latter part of the nineteenth century brought a change. The decimal point is placed high, as in 1*63, by I. Lernoch 4 of Lembcrg. N. Fialkowski of Vienna in 1863 uses the elevated dot 5 and also in 1892. 6 The same practice is followed by A. Steinhauser of Vienna, 7 by Johann Spielmann 8 and Richard Supplantschitsch, 9 and by Karl 1 Friedrich Meyer, Driller Cursus der Planimelrie (Halle a/8., 1885), p. 5. 2 P. Mako e S.I., De . . . . aequalionvm resolvlionibvs libri dvo (Vienna, 1770), p. 135; Compendiaria Malhcseos Institvtio Pavlvs Mako e S.I. in Coll. Keg. Thcres Prof. Math, et Phys. Experim. (editio tertia; Vienna, 1771). 3 Franz Scraphin Mozhnik, Theorie der numcrischen Glcichungen (Wien, 1839), p. 27, 33. 4 Ignaz Lemoch, Lchrbuch der praklischen Geomelrie, 2. Theil, 2. Aufl. (Wien, 1857), p. 163. 6 Nikolaus Fialkowski, Das Decimalrcchnen mil Rangziffern (Wien, 1863), p. 2. 6 N. Fialkowski, Praktische Geomclrie (Wien, 1892), p. 48. 7 Anton Steinhauser, Lehrbuch der Mathematik. Algebra (Wien, 1875), p. Ill, 138. 8 Johann Spielmann, Motniks Lehrbuch der Geometric (Wien, 1910), p. 66. 9 Richard Supplantschitsch, Malhemalisches Unlerrichlswerk, Lehrbuch der Geomelrie (Wien, 1910), p. 91. 334 A HISTORY OF MATHEMATICAL NOTATIONS Rosenberg. 1 Karl Zahradnfcek 2 writes 0-35679.1'0765.1'9223.0'3358, where the lower dots signify multiplication and the upper dots are decimal points. In the same way K. Wolletz 3 writes ( 0'0462). 0-0056. An isolated instance of the use of the elevated dot as decimal separatrix in Italy is found in G. Peano. 4 In France the comma placed low is the ordinary decimal separa- trix in mathematical texts. But the dot and also the comma are used in marking off digits of large numbers into periods. Thus, in a political and literary journal of Paris (1908) 6 one finds "2,251,000 drachmes," "Fr. 2.638.370 75," the francs and centimes being separated by a vacant place. One finds also "601,659 francs 05" for Fr. 601659. 05. It does not seem customary to separate the francs from centimes by a comma or dot. That no general agreement in the notation for decimal fractions exists at the present time is evident from the publication of the In- ternational Mathematical Congress in Strasbourg (1920), where deci- mals are expressed by commas 6 as in 2,5 and also by dots 7 as in 2.5. in that volume a dot, placed at the lower border of a line, is used also to indicate multiplication. 8 The opinion of an American committee of mathematicians is expressed in the following: "Owing to the frequent use of the letter x, it is preferable to use the dot (a raised period) for multiplication in the few cases in which any symbol is necessary. For example, in a case like 1-2-3 . . . . (x l)-z, the center dot is preferable to the symbol X; but in cases like 2a(x a) no symbol is necessary. The committee recognizes that the period (as in a. 6) is more nearly international than the center dot (as in a -6); but inasmuch as the period will continue to be used in this country as a decimal point, 1 Karl Rosenberg, Lehrbuch der Physik (Wien, 1913), p. 125. 2 Karl Zahradnfek, Mocniks Lehrbuch der Arithmetik und Algebra (Wien, 1911), p. 141. 3 K. Wolletz, Arithmetik und Algebra (Wien, 1917), p. 163. 4 Giuseppe Peano, Risoluzione graduate delle equazioni numeriche (Torino, 1919), p. 8. Reprint from Atti delta r. Accad. delle Scienze di Torino, Vol. LIV (1918-19). 6 Les Annales, Vol. XXVI, No. 1309 (1908), p. 22, 94. 6 Comptes rendus du congres international des mathematiques (Strasbourg, 22-30 Septembre 1920; Toulouse, 1921), p. 253, 543, 575, 581. 7 Op. cit., p. 251. < Op. tit., p. 1.53,252,545. POWERS 335 it is likely to cause confusion, to elementary pupils at least, to attempt to use it as a symbol for multiplication. " l 289. Signs for repeating decimals. In the case of repeating deci- mals, perhaps the earliest writer to use a special notation for their designation was John Marsh, 2 who, "to avoid the Trouble for the future of writing down the Given Repetend or Circulate, whether Single or Compound, more than once," distinguishes each "by placing a Period over the first Figure, or over the first and last Figures of the given Repetend." Likewise, John Robertson 3 wrote 0,3 for 0,33 . . . . , 0,23 for 0,2323 , 0,785 for 0,785785 H. Clarke 4 adopted / / / the signs .6 for .666 . . . . , .642 for .642642 A choice favoring the dot is shown by Nicolas Pike 5 who writes, 379, and by Robert Pott 6 and James Pryde 7 who write *3, '45, '34567. A return to ac- cents is seen in the Dictionary of Davies and Peck 8 who place accents over the first, or over the first and last figure, of the repetend, thus: .'2, .'5723', 2.418'. SIGNS OF POWERS 290. General remarks. An ancient symbol for squaring a number occurs in a hieratic Egyptian papyrus of the late Middle Empire, now in the Museum of Fine Arts in Moscow. 9 In the part containing the computation of the volume of a frustrated pyramid of square base there occurs a hieratic term, containing a pair of walking legs J\ and signifying "make in going," that is, squaring the number. The Diophantine notation for powers is explained in 101, the Hindu notation in 106, 110, 112, the Arabic in 116, that of Michael Psellus in 117. The additive principle in marking powers is referred 1 The Reorganization of Mathematics in Secondary Schools, by the National Com- mittee on Mathematical Requirements, under the auspices of the Mathematical Association of America (1923), p. 81. 2 John Marsh, Decimal Arithmetic Made Perfect (London, 1742), p. 5. 3 John Robertson, Philosophical Transactions (London, 1768), No. 32, p. 207- 13. See Tropfke, op. cit., Vol. I (1921), p. 147. 4 H. Clarke, The Rationale of Circulating Numbers (London, 1777), p. 15, 16. 6 Nicolas Pike, A New and Complete System of Arithmetic (Newbury-port, 1788), p. 323. 8 Robert Pott, Elementary Arithmetic, etc. (Cambridge, 1876), Sec. X, p. 8. 7 James Pryde, Algebra Theoretical and Practical (Edinburgh, 1852), p. 278. 8 C. Davies and W. G. Peck, Mathematical Dictionary (1855), art. ' 'Circulating Decimal." 9 See Ancient Egypt (1917), p. 100-102. 336 A HISTORY OF MATHEMATICAL NOTATIONS to in 101, 111, 112, 124. The multiplicative principle in marking powers is elucidated in 101, 111, 116, 135, 142. Before proceeding further, it seems desirable to direct attention to certain Arabic words used in algebra and their translations into Latin. There arose a curious discrepancy in the choice of the princi- pal unknown quantity; should it be what we call x, or should it be # 2 ? al-Khowarizmi and the older Arabs looked upon a; 2 as the principal unknown, and called it mal ("assets," "sum of money"). 1 This view- point may have come to them from India. Accordingly, x (the Arabic jidr, "plant-root," "basis," "lowest part") must be the square root of mal and is found from the equation to which the problem gives rise. By squaring x the sum of money could be ascertained. Al-Khowarizmi also had a general term for the unknown, shai ("thing"); it was interpreted broadly and could stand for either mal or jidr (z 2 or x). Later, John of Seville, Gerard of Cremona, Leonardo of Pisa, translated the Arabic jidr into the Latin radix, our x; the Arabic shai into res. John of Seville says in his arithmetic: 2 "Quaeri- tur ergo, quae res cum. X. radicibus suis idem decies acccpta radice sua efficiat 39." ("It is asked, therefore, what thing together with 10 of its roots or what is the same, ten times the root obtained from it, yields 39.") This statement yields the equation 2 +10x = 39. Later shai was also translated as causa } a word which Leonardo of Pisa used occasionally for the designation of a second unknown quantity. The Latin res was translated into the Italian word cosa, and from that evolved the German word coss and the English adjective "cossic." We have seen that the abbreviations of the words cosa arid cubus, viz., co. and cu., came to be used as algebraic symbols. The words numerus, dragma, denarius, which were often used in connection with a given absolute number, experienced contractions sometimes em- ployed as symbols. Plato of Tivoli, 3 in his translation from the Hebrew of the Liber embadorum of 1145, used a new term, latus ("side"), for the first power of the unknown, x, and the name embadum ("con- tent") for the second power, x 2 . The term latus was found mainly in early Latin writers drawing from Greek sources and was used later by Ramus ( 322), Vieta ( 327), and others. 291. Double significance of "R" and "1." There came to exist considerable confusion on the meaning of terms and symbols, not only 1 J. Ruska, Sitzungxberichte Heidelberger Akad. } PhiL-hist. Klasse (1917), Vol. II, p. 61 f.; J. Tropfke, op. ciL, Vol. II (2d ed., 1921), p. 106. 2 Tropfke, op. cit., Vol. II (2d ed., 1921), p. 107. 8 M. Curtze, Bibliotheca mathematica (3d scr.), Vol. I (1900), p. 322, n. 1. POWERS 337 because res (x) occasionally was used for x 2 , but more particularly because both radix and latus had two distinct meanings, namely, x and Vx. The determination whether x or Vx was meant in any par- ticular case depended on certain niceties of designation which the unwary was in danger of overlooking ( 137). The letter I (latus) was used by Ramus and Vieta for the designa- tion of roots. In some rare instances it also represented the first power of the unknown x. Thus, in Schoner's edition of Ramus 1 51 meant 5x, while 15 meant 1/5. Schoner marks the successive powers "/., q., c., bq., J., qc., 6.J., tq., cc." and named them latus, quadratus, cubus, biquadratus, and so on. Ramus, in his Scholarvm mathematicorvm libri unus et triginti (1569), uses the letter / only for square root, not for x or in the designation of powers of x; but he uses (p. 253) the words latus, quadratus, latus cubi for x, x z , a; 3 . This double use of I is explained by another pupil of Ramus, Bernardus Salignacus, 2 by the statement that if a number precedes the given sign it is the coefficient of the sign which stands for a power of the unknown, but if the number comes immediately after the I the root of that number is to be extracted. Accordingly, 2q, 3c, 51 stand respectively for 2x 2 , 3z 3 , 5x; on the other hand, 15, IcS, IbqW stand respectively for 1/5, ^8, V 16. The double use of the capital L is found in G. Gosselin ( 174, 175). B. Pitiscus 3 writes our 3z z 3 thus, 31! Ic, and its square 9<? G.bq+lqc, while Willebrord Snellius 4 writes our 5x 5x 3 +a: 5 in the form 51 5c+l/3. W. Oughtred 5 writes x 5 -15.r 4 +160x 3 -1250o; 2 + 6480s =170304782 in the form lqc-l5qq+lGQc-l25Qq+Q48Ql = 170304782. Both /. and R. appear as characters designating the first power 1 Petri Rami Veromandui Philosophi .... arithmetica libri duo et geometriae septem et viginti. Dudum quidem a Lazaro Schonero .... (Francofvrto ad moe- nvm, MDCXXVII). P. 139 begins: "De Nvmeris figvratis Lazari Schoneri liber." See p. 177. 2 Bernard 'i Salignaci Burdegalensis Algebrae libri duo (Francofurti, 1580). See P. Trcutlein in Abhandl. zur Geschichte der Mathematik, Vol. II (1879), p. 36. 3 Batholomaei Pitisd .... Trigonometriae editio tertia (Francofurti, 1612), p. 60. 4 Willebrord Snellius, Doctrinae triangvlorvm cononicae liber qvatvor (Leyden, 1627), p. 37. 6 William Oughtred, Clavis mathematicae, under "De aequationum affectarum resolutione in numeris" (1647 and later editions). 338 A HISTORY OF MATHEMATICAL NOTATIONS in a work of J. J. Heinlin 1 at Tubingen in 1679. He lets N stand for unitas, numerus absolutus, if , /., R. for latus vel radix; z., q. for quad- rains, zensus; ce, c for cubus; zz, qq, bq for biquadratics. But he utilizes the three signs *f,l y R also for indicating roots. He speaks 2 of "Latus cubicum, vel Radix cubica, cujus nota est Lc. R.c. if. ce." John Wallis 3 in 1655 says "Est autem lateris I, numerus pyramida- lis 1 3 +W+21" and in 1685 writes 4 "ll-2laa+a*: +bb," where the I takes the place of the modern x and the colon is a sign of aggregation, indicating that all three terms are divided by 6 2 . 292. The use of I}. (Radix) to signify root and also power is seen in Leonardo of Pisa ( 122) and in Luca Pacioli ( 136, 137). The sign /? was allowed to stand for the first power of the unknown x by Peletier in his algebra, by K. Schott 5 in 1661, who proceeds to let Q. stand for x 2 , C. for x 3 , Biqq or qq. for x 4 , Ss. for x 5 , Cq. for x 6 , SsB. for x 7 , Trig, or qqq for x 8 , Cc. for x 9 . One finds # in W. Leybourn's publication of J. Billy's 6 Algebra, where powers are designated by the capital letters N, R, Q, QQ, 8, QC, 52, QQQ., and where x 2 = 20-x is written "lQ = 20-lfi." Years later the use of R. for x and of !}. (an inverted capital letter E, rounded) for x 2 is given by Tobias Beutel 7 who writes "21 , gleich 2100, 1 (?. gleigh 100, IR. gleich 10." 293. Facsimilis of symbols in manuscripts. Some of the forms for radical signs and for x, x 2 , x 3 , x 4 , and x 5 , as found in early German manuscripts and in Widman's book, are tabulated by J. Tropfke, and we reproduce his table in Figure 104. In the Munich manuscript cosa is translated ding; the symbols in Figure 104, C2a, seem to be modified d's. The symbols in C3 are signs for res. The manuscripts C3b, C6, C7, C9, H6 bear on the evolution of the German symbol for x. Paleographers incline to the view that it is a modification of the Italian co, the o being highly disfigured. In B are given the signs for dragma or numerus. 1 Joh. Jacobi Heirdini Synopsis mathematica universalis (3d ed. ; Tubingen, 1679), p. 66. 2 Ibid., p. 65. 3 John Wallis, Arithmetica infinitorum (Oxford, 1655), p. 144. 4 John Wallis, Treatise of Algebra (London, 1685), p. 227. 6 P. Gasparis Schotti .... Cursus mathematicus (Herbipoli [Wiirtzburg], 1661), p. 530. 8 Abridgement of the Precepts of Algebra. The Fourth Part. Written in French by James Billy and now translated into English Published by Will. Ley- bourn (London, 1678), p. 194. 7 Tobias Beutel, Geomelrische Galleri (Leipzig, 1690), p. 165. POWERS 339 294. Two general plans for marking powers. In the early develop- ment of algebraic symbolism, no signs were used for the powers of given numbers in an equation. As given numbers and coefficients were not represented by letters in equations before the time of Vieta, but were specifically given in numerals, their powers could be computed on the spot and no symbolism for powers of such numbers was needed. It was different with the unknown numbers, the determination of which con- stituted the purpose of establishing an equation. In consequence, Mttnchener cod. lt 14908 (145& 1461) IT J48 Drcsdener lUndsctrlftenbana C. 80 (vor 1486) Deutacbe Algebra fol. 868878' Lateiniaehe Algebr* fol. 350-86S' Kleine Ut Algebr* fol288-288' Wiener HndcbTifln- band Kr. 62 17. (am 1MX); 7 Wl | l oel Algobrg fol. Iff. Rf 74. ft 94$ ieff KB. Cod. Dmd, 0. SO fol 289, 292' lind etwM p&tre Eintrtgnngen. FIG. 104. Signs found in German manuscripts and early German books. (Taken from J. Tropfkc, op. cit., Vol. II [2d ed., 1921], p. 112.) one finds the occurrence of symbolic representation of the unknown and its powers during a period extending over a thousand years before the introduction of the literal coefficient and its powers. For the representation of the unknown there existed two general plans. The first plan was to use some abbreviation of a name signify- ing unknown quantity and to use also abbreviations of the names signifying the square and the cube of the unknown. Often special symbols were used also for the fifth and higher powers whose orders were prime numbers. Other powers of the unknown, such as the fourth, sixth, eighth powers, were represented by combinations of those symbols. A good illustration is a symbolism of Luca Paciola, in which co. (cosa) represented x, ce. (censo) x 2 , cu. (cube) re 3 , p.r. (primo relato) x 6 ; combinations of these yielded ce.ce. for x 4 , ce.cu. for 340 A HISTORY OF MATHEMATICAL NOTATIONS x 6 , etc. We have seen these symbols also in Tartaglia and Cardan, in the Portuguese Nuftez ( 166), the Spanish Perez de Moya in 1652, and Antich Rocha 1 in 1564. We may add that outside of Italy Pacioli's symbols enjoyed their greatest popularity in Spain. To be sure, the German Marco Aurel wrote in 1552 a Spanish algebra ( 165) which contained the symbols of Rudolff, but it was Perez de Moya and Antich Rocha who set the fashion, for the sixteenth century in Spain; the Italian symbols commanded some attention there even late in the eighteenth century, as is evident from the fourteenth un- revised impression of Perez de Moya's text which appeared at Madrid in 1784. The 1784 impression gives the symbols as shown in Figure 105, and also the explanation, first given in 1562, that the printing office does not have these symbols, for which reason the ordinary letters of the alphabet will be used. 2 Figure 105 is interesting, for it purports to show the handwritten forms used by De Moya. The symbols are not the German, but are probably derived from them. In a later book, the Tratado de Mathematicas (Alcala, 1573), De Moya gives on page 432 the German symbols for the powers of the unknown, all except the first power, for which he gives the crude imitation Ze. Antich Rocha, in his Arithmetica, folio 253, is partial to capita i letters and gives the successive powers thus: N, Co, Ce, Cu, Cce, k, CeCu, RR, Ccce, Ccu, etc. The same fondness for capitals is shown i i his Mew for "more" (320). We digress further to state that the earliest mathematical work published in America, the Sumario compendioso of Juan Diez Freyle 3 1 Arithmelica por Antich Rocha de Gerona compuesta, y de varies Auctores recopilada (Barcelona, 1564, also 1565). 2 Juan Perez de Moya, Aritmetica practica, y especulativa (14th ed.; Madrid, 1784), p. 263: "Por los diez caracte"res, que en el precedente capftulo se pusieron, uso estos. Por el qual dicen numero n. por la cosa, co. por el censo, ce. por cubo, cu. por censo, de censo, cce. por el primero relato, R. por el censo, y cubo, ce.cu. por segundo relato, RR. por censo de censo de censo, cce. por cubo de cubo, ecu. Esta figura r. quiere decir raiz quadrada, Esta figura rr. denota raiz quadrada de raiz quadrada. Estas rrr. denota raiz cubica. De estos dos caract6res, p. m. notards, que la p. quiere decir mas, y la m. menos, el uno es copulativo, el otro disyuntivo, sirven para sumar, y restar cantidades diferentes, como adelante mejor eriteriderjis. Quando despues de r. se pone u. denota raiz quadrada universal: y asi rru. raiz de raiz quadrada universal: y de esta suerte rrru. raiz cubica uni- versal. Esta figura ig. quiere decir igual. Esta q. denota cantidad, y asi qs. canti- dades: estos caractc*res me ha parecido poner, porque no habia otros en la Impren- ta; tu podras usar, q nan do hagas demandas, de los que se pusicron en el segundo capitulo, porque son rnas breves, en lo dcmds todos son de una condicion." 3 Edition by D. E. Smith (Boston and London, 1921). POWERS 341 (City of Mexico, 1556) gives six pages to algebra. It contains the words cosa, zenso, or censo, but no abbreviations for them. The work does not use the signs + or , nor the p and m. It is almost purely rhetorical. The data which we have presented make it evident that in Perez de Moya, Antich Rocha, and P. Nunez the symbols of Pacioli are used and that the higher powers are indicated by the combinations of symbols of the lower powers. This general principle underlies the no- tations of Diophantus, the Hindus, the Arabs, and most of the Ger- mans and Italians before the seventeenth century. For convenience we shall call this the "Abbreviate Plan." Cap. II. En el qualsepontn algunos car after es> quesirwnpor cantidades proportionates. En este capitulo se ponen algunos cara&eres , dando a cada imo el nombre y valor que le conviene. Los quales son inveu- tados por causa de brevedad; y es de saber, que no es de nece- sidad, que estos, y no otros hayan de ser, porque cada uno pue- de usar de lo que cjuisiere , e inventar mucho mas , procedien- do con Ja proporcion que le pareciere. Los cara&eres son estos. FIG. 105. The written algebraic symbols for powers, as given in Perez de Moya/s Arithmetica (Madrid, 1784), p. 260 (1st ed., 1562). The successive sym- bols are called cosa es raiz, cen.so, cubo, censo de censo, primero relato, censo y cubo, segimdo rdato, censo de censo de censo, cubo de cubo. The second plan was not to use a symbol for the unknown quantity itself, but to limit one's self in some way to simply indicating by a numeral the power of the unknown quantity. As long as powers of only one unknown quantity appeared in an equation, the writing of the index of its power was sufficient. In marking the first, second, third, etc., powers, only the numerals for "one," "two," "three," etc., were written down. A good illustration of this procedure is Chuquet's 10 2 for lOz 2 , 10 1 for Wx, and 10 for 10. We shall call this the "Index Plan." It was stressed by Chuquet, and passed through several stages of development in Bombelli, Stevin, and Girard. Then, after the introduction of special letters to designate one or more un- known quantities, and the use of literal coefficients, this notation was perfected by Hrigone and Hume; it finally culminated in the present- day form in the writings of Descartes, Wallis, and Newton. 342 A HISTORY OF MATHEMATICAL NOTATIONS 295. Early symbolisms. In elaborating the notations of powers according to the "Abbreviate Plan" cited in 294, one or the other of two distinct principles was brought into play in combining the symbols of the lower powers to mark the higher powers. One was the additive principle of the Greeks in combining powers; the other was the multiplicative principle of the Hindus. Diophantus expressed the fifth power of the unknown by writing the symbols for x 2 and for y? y one following the other; the indices 2 and 3 were added. Now, Bhaskara writes his symbols for x 2 and y? in the same way, but lets the two designate, not s 5 , but rr 6 ; the indices 2 and 3 are multiplied. This difference in designation prevailed through the Arabic period, the later Middle Ages in Europe down into the seventeenth century. It disappeared only when the notations of powers according to the "Abbreviate Plan" passed into disuse. References to the early sym- bolisms, mainly as exhibited in our accounts of individual authors, are as follows: ABBREVIATE PLAN ADDITIVE PRINCIPLE Diophantus, and his editors Xylander, Bachet, Fermat ( 101) al-Karkhi, eleventh century (116) Leonardo of Pisa ( 122) Anonymous Arab ( 124) Dresden Codex C. 80 ( 305, Fig. 104) M. Stifel, (1545), sum; sum; sum: x* ( 154) F. Vieta (1591), and in later publications ( 177) C. Glorioso, 1527 ( 196) W. Oughtred, 1631 ( 182) Samuel Foster, 1659 ( 306) MULTIPLICATIVE PRINCIPLE Bhaskara, twelfth century ( 110-12) Arabic writers, except al-Karkhi ( 116) L. Pacioli, 1494, ce. cu. for z 6 ( 136) H. Cardano, 1539, 1545 ( 140) N. Tartaglia, 1556-60 ( 142) Ch. Rudolff, 1525 ( 148) M. Stifel, 1544 ( 151) J. Scheubel, 1551, follows Stifel ( 159) A. Rocha, 1565, follows Pacioli ( 294) C. Clavius, 1608, follows Stifel ( 161) P. Nufiez, 1567, follows Pacioli and Cardan ( 166) R. Recorde, 1557, follows Stifel ( 168) POWERS 343 L. and T. Digges, 1579 ( 170) A. M. Visconti, 1581 ( 145) Th. Masterson, 1592 ( 171) J. Peletier, 1554 ( 172) G. Gosselin, 1577 ( 174) L. Schemer, 1627 (291) NEW NOTATIONS ADOPTED Ghaligai and G. del Sodo, 1521 ( 139) M. Stifel, 1553, repeating factors ( 156) J. Buteon, 1559 ( 173) J. Scheubel, N, Ra, Pri, Se ( 159) Th. Harriot, repeating factors ( 188) Johann Geysius, repeating factors ( 196, 305) John Newton, 1654 (305) Nathaniel Torporley ( 305) Joseph Raphson, 1702 ( 305) Samuel Foster, 1659, use of lines ( 300) INDEX PLAN Psellus, nomenclature without signs ( 117) Neophytos, scholia ( 87, 88) Nicole Oresine, notation for fractional powers ( 123) N. Chuquet, 1484, 12 3 for 12r* ( 131) E. de la Roche, 1520 ( 132) R. Bombelli, 1572 ( 144) Grammateus, 1518, pri, se., ter. quart. ( 147) G. van der Hoecke, 1537, pri, se, 3 a ( 150) S. Stevin ( 162) A. Girard, 1629 ( 164) L. & T. Digges, 1579 ( 170, Fig. 76) P. HSrigone, 1634 ( 189) J. Hume, 1635, 1636 ( 190) 296. Notations applied only to an unknown quantity, the base being omitted. As early as the fourteenth century, Oresme had the ex- ponential concept, but his notation stands in historical isolation and does not constitute a part of the course of evolution of our modern exponential symbolism. We have seen that the earliest important steps toward the modern notation were taken by the Frenchman Nicolas Chuquet, the Italian Rafael Bombelli, the Belgian Simon Stevin, the Englishmen L. and T. Digges. Attention remains to be called to a symbolism very similar to that of the Digges, which was contrived by Pietro Antonio Cataldi of Bologna, in an algebra of 1610 344 A HISTORY OF MATHEMATICAL NOTATIONS and a book on roots of 1613. Cataldi wrote the numeral exponents in their natural upright position, 1 and distinguished them by crossing them out. His "5 3 via 8 if a 40 7" means 5z 3 8# 4 =40o; 7 . His sign for x is Z. He made only very limited use of this notation. The drawback of Stevin's symbolism lay in the difficulty of writing and printing numerals and fractions within the circle. Apparently as a relief from this cumbrousness, we find that the Dutch writer, Adrianus Romanus, in his Ideae Mathematicae pars prima (Antwerp, 1593), uses in place of the circle two rounded parentheses and vinculums above and below; thus, with him 1(45) stands for z 45 . He uses this notation in writing his famous equation of the forty-fifth degree. Franciscus van Schooten 2 in his early publications and when he quotes from Girard uses the notation of Stevin. A notation more in line with Chuquet's was that of the Swiss Joost Biirgi who, in a manuscript now kept in the library of the ob- servatory at Pulkowa, used Roman numerals for exponents and wrote 3 8+12-9+10+3+7-4 for 8z 6 +12z 5 -9z 4 +10^+3z 2 +7:r--4 . In this notation Biirgi was followed by Nicolaus Reymers (1601) and J, Kepler. 4 Reymers 5 used also the cossic symbols, but chose R in place of 7; occasionally he used a symbolism as in 25IIII+20II 10III-8I for the modern 25z 4 +20r 2 - 10z 3 -8z. We see that Cataldi, Romanus, Fr. van Schooten, Biirgi, Reymers, and Kepler belong in the list of those who followed the "Index Plan." 297. Notations applied to any quantity, the base being designated. As long as literal coefficients were not used and numbers were not generally represented by letters, the notations of Chuquet, Bombelli, 1 G. Wertheim, Zeitschr. f. Math. u. Physik, Vol. XLIV (1899), Hist.-Lit. Abteilung, p. 48. 2 Francisci a Schooten, De Organica conicarum sectionum .... Tractatus (Leyden, 1646), p. 96; Schooten, Renali Descartes Geometria (Frankfurt a./M M 1695), p. 359. 3 P. Treutlein in Abhandlungen zur Oeschichte der Mathemalik, Vol. II (Leipzig, 1879), p. 36, 104. 4 In his "De Figurarum regularium" in Opera omnia (ed. Ch, Frisch), Vol. V (1864), p. 104, Kepler lets the radius AB of a circle be 1 and the side BC of a regular inscribed heptagon be R. He says: "In hac proportione continuitatem fingit, ut sicut est ABl ad BC IR, sic sit IR ad Iz, et 10 as 1 c, et 1 c ad Izz, et Izzad Iz < et sic perpetuo, quod nos commodius signabimus per apices six, 1, I 1 , 1", 1"', 1 IV , F, ivi ; IVH, etc." 6 N. Raimarus Ursus, Arithmetica analytica (Frankfurt a. O., 1601), Bl. C3v. See J. Tropfke, op. tit., Vol. II (2d ed., 1921), p. 122. POWERS 345 Stevin, and others were quite adequate. There was no pressing need of indicating the powers of a given number, say the cube of twelve; they could be computed at once. Moreover, as only the unknown quantity was raised to powers which could not be computed on the spot, why should one go to the trouble of writing down the base? Was it not sufficient to put down the exponent and omit the base? v Was it not easier to write 16 than IGx 5 ? But when through the inno- vations of Vieta and others, literal coefficients came to be employed, and when several unknowns or variables came to be used as in ana- lytic geometry, then the omission of the base became a serious defect ii ii in the symbolism. It will not do to write 15z 2 16?/ 2 as 1516. In watching the coming changes in notation, the reader will bear this problem in mind. Vieta/s own notation of 1591 was clumsy: D quadra- turn or D. quad, stood for D 2 , D cubum for D 8 ; A quadr. for z 2 , A repre- senting the unknown number. In this connection perhaps the first writer to be mentioned is Luca Pacioli who in 1494 explained, as an alternative notation of powers, the use of R as a base, but in place of the exponent he employs an ordinal that is too large by unity ( 136). Thus R. 30* stood for a: 29 . Evidently Pacioli did not have a grasp of the exponential concept. An important step was taken by Romanus 1 who uses letters and writes bases as well as the exponents in expressions like A (4) +(4) +4^1(3) in +6A(2) in B(2)+A in B(3) which signifies A similar suggestion came from the Frenchman, Pierre H6rigone, a mathematician who had a passion for new notations. He wrote our a 3 as a3, our 26 4 as 264, and our 2fea 2 as 2ba2. The coefficient was placed before the letter, the exponent after. In 1636 James Hume 2 brought out an edition of the algebra of Vieta, in which he introduced a superior notation, writing down the base and elevating the exponent to a position above the regular line and a little to the right. The exponent was expressed in Roman 1 See H. Bosnians in Annales Societe sclent, de Bruxelles, Vol. XXX, Part II (1906), p. 15. 2 James Hume, L'Algebre de Viete, d'une methode nouvelle claire et facile (Paris 1636). See (Euvres de Descartes (ed. Charles Adam et P. Tannery), Vol. V, p. 604, 506-12. 346 A HISTORY OF MATHEMATICAL NOTATIONS numerals. Thus, he wrote A Jii for A 3 . Except for the use of the Ro- man numerals, one has here our modern notation. Thus, this Scots- man, residing in Paris, had almost hit upon the exponential symbolism which has become universal through the writings of Descartes. 298. Descartes 9 notation of 1637. Thus far had the notation ad- vanced before Descartes published his Geometric (1637) ( 191). Hdrigone and Hume almost hit upon the scheme of Descartes. The only difference was, in one case, the position of the exponent, and, in the other, the exponent written in Roman numerals. Descartes ex- pressed the exponent in Arabic numerals and assigned it an elevated position. Where Hume would write 5a iv and H6rigone would write 5a4, Descartes wrote 5a 4 . From the standpoint of the printer, H6ri- gone's notation was the simplest. But Descartes' elevated exponent offered certain advantages in interpretation which the judgment of subsequent centuries has sustained. Descartes used positive integral exponents only. 299. Did Stampioen arrive at Descartes' notation independently? Was Descartes alone in adopting the notation 5a 4 or did others hit upon this particular form independently? In 1639 this special form was suggested by a young Dutch writer, Johan Stampioen. 1 He makes no acknowledgment of indebtedness to Descartes. He makes it ap- pear that he had been considering the two forms 3 a and a 3 , and had found the latter preferable. 2 Evidently, the symbolism a 3 was adopted by Stampioen after the book had been written; in the body of his book 3 one finds aaa, bbbb, fffff, gggggg, but the exponential notation above noted, as described in his passage following the Preface, is not used. Stampioen uses the notation a 4 in some but not all parts of a controversial publication 4 of 1640, on the solution of cubic equations, and directed against Waessenaer, a personal friend of Descartes. In view of the fact that Stampioen does not state the originators of any of the notations which he uses, it is not improbable that his a 3 was taken from Descartes, even though Stampioen stands out as an opponent of Descartes. 5 1 Johan Stampioen d'Jonghe, Algebra ofte Nieuwe Stel-Regel (The Hague, 1639). See his statement following the Preface. 2 Stampioen's own words are: "aw* dit is a drievoudich in hem selfs gemen- nichvuldicht. men soude oock daer voor konnen stellen 8 a ofte better aV 8 J. Stampioen, op. tit., p. 343, 344, 348. 4 7. /. Stampioenii Wis-Konstigh ende Reden-Maetigh Bewys ('s Graven- Hage, 1640), unpaged Introduction and p. 52-55. 6 (Euvres de Descartes, Vol. XII (1910), p. 32, 272-74. POWERS 347 300. Notations used by Descartes before 1637. Descartes' indebt- edness to his predecessors for the exponential notation has been noted. The new features in Descartes' notation, 5a 3 , 6a6 4 , were in- deed very slight. What notations did Descartes himself employ before 1637? In his Opuscules de 1619-1621 he regularly uses German symbols as they are found in the algebra of Clavius; Descartes writes 1 "36~32-67ea65u.lt/' which means 36 3# 2 6# = :c 3 . These Opuscules were printed by Toucher de Careil (Paris, 1859-60), but this printed edition contains corruptions in notation, due to the want of proper type. Thus the numeral 4 is made to stand for the German symbol J^ ; the small letter 7 is made to stand for the radical sign j/ The various deviations from the regular forms of the symbols are set forth in the standard edition of Descartes' works. Elsewhere ( 264) we call attention that Descartes 2 in a letter of 1640 used the Recordian sign of equality and the symbols N and C of Xy lander, in the expression "1C QN = 40." Writing to Mersenne, on May 3, 1638, Descartes 3 employed the notation of Vieta, "Aq+Bq+A in B bis" for our a 2 +& 2 +2a6. In a posthumous document, 4 of which the date of composition is not known, Descartes used the sign of equality found in his Geomitrie of 1637, and P. Hrigone's notation for powers of given letters, as b3x for b*x, a3z for a?z. Probably this document was written before 1637. Descartes 5 used once also the notation of Dounot (or Deidier, or Bar-le-Duc, as he signs himself in his books) in writing the equation lC-$Q+13Neq. 1/288-15, but Descartes translates it into t/ 3 -9y 2 + 13!/- 121/2+ 15 ^0 0. 301. Use of Herigone's notation after 1 637. After 1637 there was during the seventeenth century still very great diversity in the ex- ponential notation. H6rigone's symbolism found favor with some writers. It occurs in Florimond Debeaune's letter 6 of September 25, 1638, to Mersenne in terms like 2?/4, 2/3, 2Z2 for 2^, i/ 3 , and 2P, re- 1 Ibid., Vol. X (1908), p. 249-51. See also E. de Jonquieres in Bibliotheca mathematica (2d ser.), Vol. IV (1890), p. 52, also G. Enestrom, Bibliotheca mathe- matwa (3d ser.), Vol. VI (1905), p. 406. 2 (Euvres de Descartes, Vol. Ill (1899), p. 190. *Ibid., Vol. II (1898), p. 125; also Vol. XII, p. 279. 4 Ibid., Vol. X (1908), p. 299. 5 Ibid., Vol. XII, p. 278. 9 Ibid., Vol. V (1903), p. 516. 348 A HISTORY OF MATHEMATICAL NOTATIONS spectively. G. Schott 1 gives it along with older notations. Pietro Mengoli 2 uses it in expressions like a4+4a3r+6a2r2+4ar3+r4 for our a 4 +4a s r+6o 2 r 2 +4ar 3 +r 4 . The Italian Cardinal Michelangelo Ricci 3 writes "AC Z in CB 3 " for 1~C 2 .(JB 3 . In a letter 4 addressed to Ozanam one finds 64+c4^a4 for 6 4 +c 4 = a 4 . Chr. Huygens 5 in a letter of June 8, 1684, wrote a3+aab for a 8 +a 2 6. In the same year an article by John Craig 6 in the Philosophical Transactions contains a3y+aA for a*y+a* t but a note to the "Benevole Lector" appears at the end apolo- gizing for this notation. Dechales 7 used in 1674 and again in 1690 (along with older notations) the form A4:+4A3B+QA2B2+4:AB3 + 54. A Swedish author, Andreas Spole, 8 who in 1664-66 sojourned in Paris, wrote in 1692 an arithmetic containing expressions 3a3+ 3a2-2a-2 for 3a 3 +3a 2 -2a-2. Joseph Moxon 9 lets "A-B.(2)" stand for our (A~B)\ also "A-B.(3)" for our (A-B)*. With the eighteenth century this notation disappeared. 302. Later use of Hume's notation of 1636. Hume's notation of 1636 was followed in 1638 by Jean de Beaugrand 10 who in an anony- mous letter to Mersenne criticized Descartes and states that the equa- tion z lv +4s ln - 19s 11 -106s- 120 has the roots +5, -2, -3, -4. Beaugrand also refers to Vieta and used vowels for the unknowns, as in "A'"+3AAB+ADP esgale a ZS8." Again Beaugrand writes "E'"<=> -13E-12" for ^-13^-12, where the o apparently desig- nates the omission of the second term, as does )(- with Descartes. 303. Other exponential notations suggested after 1637. At the time of Descartes and the century following several other exponential notations were suggested which seem odd to us and which serve to 1 G. Schott, Cursus mathematicus (Wiirzburg, 1661), p. 576. 2 Ad Maiorem Dei Gloriam Geometriae speciosae Elementa, .... Pctri Mengoli (Bologna, 1659), p. 20. 3 Michaelis Angeli Riccii Exercitatio geometrica (Londini, 1668), p. 2. [Preface, 1666.] 4 Journal des 8f avans, l'anne*e 1680 (Amsterdam, 1682), p. 160. 6 Ibid., Tanned 1684, Vol. II (2d ed.; Amsterdam, 1709), p. 254. 6 Philosophical Transactions, Vol. XV-XVI (London, 1684-91), p. 189. 7 R. P. Claudii Francisci Milliet Dechales Camberiensis Mundus rnathematicus, Tomus tertius (Leyden, 1674), p. 664; Tomus primus (editio altera; Leyden, 1690), p. 635. 8 Andreas Spole, Arilhmelica vutgaris et specioza (Upsala, 1692). See G. Ene- strom in L'Inlermediaire des mathtmaticiens, Vol. IV (1897), p. 60. 9 Joseph Moxon, Mathematical Dictionary (London, 1701), p. 190, 191. 10 (Euvres de Descartes, Vol. V (1903), p. 506, 507. POWERS 349 indicate how the science might have been retarded in its progress under the handicap of cumbrous notations, had such wise leadership as that of Descartes, Wallis, and Newton not been available. Rich. Balarn 1 in 1653 explains a device of his own, as follows: "(2) j 3 \ , the Duplicat, or Square of 3, that is, 3X3; (4) 2 , the Quaclruplicat of 2, that is, 2X2X2X2-16." The Dutch J. Stampioen 2 in 1639 wrote DA for A 2 ; as early as 1575 F. Maurolycus 3 used D to designate the square of a line. Similarly, an Austrian, Johannes Cararnuel, 4 in 1670 gives "Q25. est Quadratum Numeri 25. hoc est, 625." Huygens 5 wrote "1000(3)10" for 1,000 = 10 3 , and "1024(10)2" for 1024 ==2 10 . A Leibnizian symbolism 6 explained in 1710 indicates the cube of AB+BC thus: [3] (AB+BC); in fact, before this time, in 1695 Leibniz 7 wrote \m\ y+a for (y+a) m . 304. Descartes preferred the notation aa to a 2 . Fr. van Schooten, 8 in 1646, followed Descartes even in writing <jq y xx rather than </ 2 , x 2 , but in his 1649 Latin edition of Descartes' geometry he wrote prefer- ably x 2 . The symbolism xx was used not only by Descartes, but also by Huygens, Hahn, Kersey, Wallis, Newton, Halley, Rolle, Euler in fact, by most writers of the second half of the seventeenth and of the eighteenth centuries. Later, Gauss 9 was in the habit of writing xx, and he defended his practice by the statement that x 2 did not take up less space than xx, hence did not fulfil the main object of a symbol. The x* was preferred by Leibniz, Ozanam, David Gregory, and Pascal. 305. The reader should be reminded at this time that the repre- sentation of positive integral powers by the repetition of the factors was suggested very early (about 1480) in the Dresden Codex C. 80 under the heading Algorithmus de additis et minutis where x 2 z and x w zzzzz; it was elaborated more systematically in 1553 by M. 1 Rich. Balam, Algebra, or The Doctrine of Composing, Inferring, and tie- solving an Equation (London, 1653), p. 9. 2 Johan Stampioen, Algebra (The Hague, 1639), p. 38. 3 D. Francisc' Mavrolyci Abbatis measancnsis Opuscula Malhcmatica (Venice, 1575) (Euclid, Book XIII),- p. 107. 4 Joannis Caramvelis Malhesis Biceps. Veins et Nova (Companiae, 1070), p. 131, 132. 5 Christ iani Hugenii Opera .... quae collegit .... Guilielmus Jacobus's Gravesande (Ley den, 1751), p. 456. Miscellanea Berolinensia (Berlin, 1710), p. 157. 7 Acta eruditorum (1695), p. 312. 8 Francisci a Schooten Leydensis de Organica conicarum sectionum .... Trac- talus (Leyden, 1646), p. 91 ff. 9 M. Cantor, op. cit., Vol. II (2d ed.), p. 794 n. 350 A HISTORY OF MATHEMATICAL NOTATIONS Stifel ( 156). One sees in Stifcl the exponential notation applied, not to the unknown but to several different quantities, all of them known. Stifel understood that a quantity with the exponent zero had the value 1. But this notation was merely a suggestion which Stifel himself did not use further. Later, in Alsted's Encyclopaedia, 1 published at Herborn in Prussia, there is given an explanation of the German symbols for radix, zensus, cubus, etc. ; then the symbols from Stifel, just referred to, are reproduced, with the remark that they are preferred by some writers. The algebra proper in the Encyclopaedia is from the pen of Johann Geysius 2 who describes a similar notation 2a, 4aa, 8aaa, . . . . , 512aaaaaaaaa and suggests also the use of I II III IX Roman numerals as indices, as in 21 4q Sc . . . . 512cc. Forty years after, Caramvel 3 ascribes to Geysius the notation aaa for the cube of a, etc. In England the repetition of factors for the designation of powers was employed regularly in Thomas Harriot. In a manuscript pre- served in the library of Sion College, Nathaniel Torporley (1573- 1632) makes strictures on Harriot's book, but he uses Harriot's notation. 4 John Newton 5 in 1654 writes aaaaa. John Collins writes in the Philosophical Transactions of 1668 aaa -3aa+4a = jV to signify a; 3 3z 2 +4:c = jV. Harriot's mode of representation is found again in the Transactions 6 for 1684. Joseph Raphson 7 uses powers of g up to 10 , but in every instance he writes out each of the factors, after the manner of Harriot. 306. The following curious symbolism was designed in 1659 by Samuel Foster 8 of London : ~1 -I =1 ID =1 3 3 ID q c qq qc cc qqc qcc ccc 2 345 6789 1 Johannis-Henrici Alstedii Encyclopaedia (Herborn, 1630), Book XIV, "Arithmetica, " p. 844. 2 Ibid., p. 865-74. 3 Joannis Caramvelis Mathesis Biceps (Campaniae, 1670), p. 121. 4 J. O. Halliwcll, A Collection of Letters Illustrative of the Progress of Science in England (London, 1841), p. 109-16. 5 John Newton, Institutio Mathematica or a Mathematical Institution (London, 1654), p. 85. 6 Philosophical Transactions, Vol. XV-XVI (London, 1684-91), p. 247, 340. 7 Josepho Raphson, Analysis Aequationum universalis (London, 1702). [First edition, 1697.] 8 Samuel Foster, Miscellanies, or Mathematical Lucubrations (London, 1659), !>. 10. POWERS 351 Foster did not make much use of it in his book. He writes the pro- portion "At AC . AR::CD\:'RP\ 9 99 which means An altogether different and unique procedure is encountered in the Maandelykse Mathematische Liefhebberye (1754-69), where \/ - signifies extracting the mth root, and - \/ signifies raising to the rath power. Thus, 307. Spread of Descartes' notation. Since Descartes' Geometric appeared in Holland, it is not strange that the exponential notation met with prompter acceptance in Holland than elsewhere. We have already seen that J. Stampioen used this notation in 1639 and 1640. The great disciple of Descartes, Fr. van Schooten, used it in 1646, and in 1649 in his Latin edition of Descartes' geometry. In 1646 van Schooten indulges 1 in the unusual practice of raising some (but not all) of his coefficients to the height of exponents. He writes # 3 *aax 2 a 3 to designate a; 3 3a 2 2a 3 = 0. Van Schooten 2 does the same thing in 1657, when he writes 2 ax for 2ax. Before this Marini Ghetaldi 3 in Italy wrote coefficients in a low position, as subscripts, as in the proportion, " ut AQ ad A 2 in B ita - ad m<>-\-n ," y Cf which stands for A*:2AB= :(2m+ri). Before this Albert Girard 4 placed the coefficients where we now write our exponents. I quote: "Soit un binome conjoint B+C. Son Cube era B(B q +C$) + )." Here the cube of B+C is given in the form corresponding 1 Francisci a Schooten, De organica conicarum sedionum .... tractatus (Leyden, 1646), p. 105. 2 Francisci a Schooten, Exerdtationum mathematicarum liber primus (Leyden, 1657), p. 227, 274, 428, 467, 481, 483. 3 Marini Ghetaldi, De resolutions et compositione mathematica libri quinque. Opus posthumum (Rome, 1630). Taken from E. Gelcich, Abhandlungen zur Ge schichte der Mathematik, Vol. IV (1882), p. 198. 4 A. Girard, Invention nouvelle en I'algebre (1629), "3 C." 352 A HISTORY OF MATHEMATICAL NOTATIONS to( 2 +3C 2 )+C(3 2 +C 2 ). Much later, in 1679, we find in the col- lected works of P. Fermat 1 the coefficients in an elevated position: 2 D in A for 2DA, 2 R in E for 2RE. The Cartesian notation was used by C. Huygens and P. Mersenne in 1646 in their correspondence with each other, 2 by J. Hudde 3 in 1658, and by other writers. In England, J. Wallis 4 was one of the earliest writers to use Descartes' exponential symbolism. He used it in 1655, even though he himself had been trained in Oughtred's notation. The Cartesian notation is found in the algebraic parts of Isaac Barrow's 5 geometric lectures of 1670 and in John Kersey's Algebra^ of 1673. The adoption of Descartes 7 a 4 in strictly algebraic operations and the retention of the older A q , A c for A 2 , A 3 in geometric analysis is of frequent occurrence in Barrow and in other writers. Seemingly, the impression prevailed that A 2 and A 3 suggest to the pupil the purely arithmetical process of multiplication, A A and A A A, but that the symbolisms A q and A c conveyed the idea of a geometric square and geometric cube. So we find in geometrical expositions the use of the latter notation long after it had disappeared from purely algebraic processes. We find it, for instance, in W. Whiston's edition of Tacquet's Euclid) 7 in Sir Isaac Newton's Principia 8 and Opticks* in B. Robins 7 Tracts, 10 and in a text by K. F. Hauber. 11 In the Philosophical Transactions of London none of the pre-Cartesian notations for powers appear, except a few times in an article of 1714 from the pen of K. Cotes, and an occasional tendency to adhere to the primitive, but very 1 P. Fermat, Varia opera (Toulouse, 1679), p. 5. 2 C. Iluygens, (Euvres, Vol. I (La Haye, 1888), p. 24. 3 Joh. Huddeni Epist. I de reductione aequatwnum (Amsterdam, 1658); Matthicsscn, Grundzuge der Antiken u. Modcrnen Algebra (Leipzig, 1878), p. 349. 4 John Wallis, Arilhmetica infinitorurn (Oxford, 1655), p. 16 ff. 5 Isaac Barrow, Lectiones Geometriae (London, 1670), Lecture XIII (W. Whcwell's ed.), p. 309. 6 John Kersey, Algebra (London, 1673), p. 11. 7 See, for instance, Elernenta Euclidea geomelriae auctore Andrea Tacquet, ... Gulielmus Whiston (Amsterdam, 1725), p. 41. 8 Sir Isaac Newton, Principia (1687), Book I, Lemma xi, Cas. 1, and in other places. 9 Sir Isaac Newton, Opticks (3d ed.; London, 1721), p. 30. 10 Benjamin Robins, Mathematical Tracts (ed. James Wilson, 1761), Vol. II, p. 65. 11 Karl Friderich Hauber, Archimeds zwey Biicher ilber Kugel und Cylinder (Tubingen, 1798), p. 56 ff. POWERS 353 lucid method of repeating the factors, as aaa for a 3 . The modern exponents did not appear in any of the numerous editions of William Oughtrcd's Clavis mathematicae; the last edition of that popular book was issued in 1694 and received a new impression in 1702. On Febru- ary 5, 1666-67, J. Wallis 1 wrote to J. Collins, when a proposed new edition of Oughtred's Clavis was under discussion: "It is true, that as in other things so in mathematics, fashions will daily alter, and that which Mr. Oughtred designed by great letters may be now by others designed by small; but a mathematician will, with the same ease and advantage, understand A c or aaa." As late as 1790 the Portuguese J. A. da Cunha 2 occasionally wrote A q and A c . J. Pell wrote r 2 and t 2 in a letter written in Amsterdam on August 7, 1645. 3 J. H. Rahn's Teutsche Algebra, printed in 1659 in Zurich, contains for positive inte- gral powers two notations, one using the Cartesian exponents, a 8 , x 4 , the other consisting of writing an Archemidean spiral (Fig. 96) be- tween the base and the exponent on the right. Thus a 3 signifies a 8 . This symbol is used to signify involution, a process which Rahn calls involviren. In the English translation, made by T. Brancker and published in 1668 in London, the Archimedean spiral is displaced by the omicron-sigrna (Fig. 97), a symbol found among several English writers of textbooks, as, for instance, J. Ward, 4 E. Hatton, 5 Ham- mond, 6 C. Mason, 7 and by P. Ronayne* all of whom use also llahn's and Brancker 's ww' to signify evolution. The omicrori-sigma is found in Birks; 9 it is mentioned by Saverien, 10 who objects to it as being superfluous. Of interest is the following passage in Newton's Arithmetick, 11 which consists of lectures delivered by him at Cambridge in the period 1669-85 and first printed in 1707: "Thus ^64 denotes 8; and i/3:64 1 Rigaud, Correspondence of Scientific Men of the Seventeenth Century, Vol. I (Oxford, 1841), p. 63. 2 J. A. da Cunha, Principios mathematicos (1790), p. 158. 3 J. O. Halliwell, Progress of Science in England (London, 1841), p. 89. 4 John Ward, The Young Mathematician's Guide (London, 1707), p. 144. 6 Edward Hatton, Inlire System of Arithmetic (London, 1721), p. 287. 6 Nathaniel Hammond, Elements of Algebra (London, 1742). 7 C. Mason in the Diarian Repository (London, 1774), p. 187. 8 Philip Ronayne, Treatise of Algebra (London, 1727), p. 3. 9 Anthony and John Birks, Arithmetical Collections (London, 1766), p. viii. 10 A. Saverien, Dictionnaire universel de mathematique et de physique (Paris, 1753), "Caractere." 11 Newton's Universal Arithmetick (London, 1728), p. 7. 354 A HISTORY OF MATHEMATICAL NOTATIONS denotes 4 There are some that to denote the Square of the first Power, make use of <?, and of c for the Cube, qq for the Biquadrate, and qc for the Quadrato-Cube, etc Others make use of other sorts of Notes, but they are now almost out of Fashion." In the eighteenth century in England, when parentheses were seldom used and the vinculum was at the zenith of its popularity, bars were drawn horizontally and allowed to bend into a vertical " IT stroke 1 (or else were connected with a vertical stroke), as in AXB U \ or in a"+"b|n"' In France the Cartesian exponential notation was not adopted as early as one might have expected. In J. de Billy's Nova geometriae clavis (Paris, 1643), there is no trace of that notation; the equation z+z 2 = 20 is written "Ifl+lQ aequatur 20." In Fermat's edition 2 of Diophantus of 1670 one finds in the introduction 1QQ+4C+10Q+ 2(W+1 for xt+4x*+10x*+20x+l. But in an edition of the works of Fermat, brought out in 1679, after his death, the algebraic notation of Vieta which he had followed was discarded in favor of the expo- nents of Descartes. 3 B. Pascal 4 made free use of positive integral exponents in several of his papers, particularly the Potestatum numeri- carum summa (1654). In Italy, C. Renaldini 5 in 1665 uses both old and new exponential notations, with the latter predominating. 308. Negative, fractional, and literal exponents. Negative and fractional exponential notations had been suggested by Oresme, Chuquet, Stevin, and others, but the modern symbolism for these is due to Wallis and Newton. Wallis 6 in 1656 used positive integral exponents and speaks of negative and fractional "indices," but he does not actually write a~ l for -, a 3 for >/a 3 . He speaks of the series 1 See, for instance, A. Malcolm, A New System of Arithmetick (London, 1730), p. 143. 2 Diophanti Alexandrini arithmeticarum Libri Sex, cum commentariis G. B. Bacheti V. C. et observationibus D. P. de Fermat (Tolosae, 1670), p. 27. 3 See (Euvres de Fermat (e*d. Paul Tannery et Charles Henry), Tome I (Paris,' 1891), p. 91 n. 4 (Euvres de Pascal (e"d. Leon Brunschvicg et Pierre Boutroux), Vol. Ill (Paris, 1908), p. 349-58. 5 Caroli Renaldinii, Ars analytica .... (Florence, 1665), p. 11, 80, 144. 6 J. Wallis, Arithmetica infinitorum (1656), p. 80, Prop. CVI. POWERS 355 _7=> _^^ e ^ c ^ as having the "index |." Our modern notation F 1 1/2 "K 3 involving fractional and negative exponents was formally introduced a dozen years later by Newton 1 in a letter of June 13, 1676, to Olden- burg, then secretary of the Royal Society of London, which explains the use of negative and fractional exponents in the statement, "Since algebraists write a 2 , a 3 , a 4 , etc., for aa, aaa, aaaa, etc., so I write a*, a 3 , a 4 , for I/a, I/a 3 , 1/c a 5 ; and I write a~ l , a~ 2 , a~ 3 , etc., for -, - , , etc." He exhibits the general exponents in his binomial formula first announced in that letter: F+PQl- -PM AQ+ m ~- n - BQ+- 2 ? CQ+-*? DQ+ , etc., H 11 o/l 4?1 m m m m where A=P n , B = - P n Q } etc., and where may represent any real it ill and rational number. It should be observed that Newton wrote here literal exponents such as had been used a few times by Wallis, 2 in 1657, in expressions like i/ d R d = R, AR m XAR n = A 2 R m + n , which arose in the treatment of geometric progression. Wallis gives also the division AR m ) AR m + n (R n . Newton 3 employs irrational exponents in his letter to Oldenburg of the date October 24, 1676, where he writes x^ 2 +x v/7 \ a =y. Before Wallis and Newton, Victa indicated general exponents a few times in a manner almost rhetorical ; 4 his A , , , E potestate A potesta . . , A potestas-] ^ , -.- A - , in A gradum ^ E gradui+A gradu J is our the two distinct general powers being indicated by the words potestas and gradus. Johann Bernoulli 5 in 1691-92 still wrote 3D frax+xx for 1 Isaaci Newtoni Opera (cd. S. Horsely), Tom. IV (London, 1782), p. 215. 2 J. Wallis, Mathesis universalis (Oxford, 1657), p. 292, 293, 294. 3 See J. Collins, Commercium epistolicum (ed. J. B. Biot and F. Lefort; Paris, 1856), p. 145. 4 Vieta, Opera mathematica (ed. Fr. van Schootcn, 1634), p. 197. 5 lohannis I Bernoulli, Lecliones de calculo differentialium .... von Paul Schafheitlin. Scparatabdruck aus den V erhandlungen der Naturforschenden Ge- sellschaft in Basel, Vol. XXXIV, 1922. 356 A HISTORY OF MATHEMATICAL NOTATIONS iCi/yx+xx for 4v // (t/x+x 2 ) 3 , 5QQl / ayx+x 3 +zyx for But fractional, negative, and general exponents were freely used by D. Gregory 1 and were fully explained by W. Jones 2 and by C. Reyneau. 3 Reyneau remarks that this theory is not ex- plained in works on algebra. 309. Imaginary exponents. The further step of introducing im- aginary exponents is taken by L. Euler in a letter to Johann Ber- noulli, 4 of October 18, 1740, in which he announces the discovery of the formula c +x|/ ~ 1 +e~ x / ~ 1 = 2 cos x, and in a letter to C. Gold- bach, 6 of December 9, 1741, in which he points out as a curiosity - that the fraction - - - is nearly equal to } J. The first ap- z pearance of imaginary exponents in print is in an article by Euler in the Miscellanea Berolinensia of 1743 and in Eulcr's Introduetio in analysin (Lausannae, 1747), Volume I, page 104, where he gives the all-important formula e +1)|// ~ 1 = cos v+l / 1 sin v. 310. At an earlier date occurred the introduction of variable exponents. In a letter of 1679, addressed to C. Huygens, G. W. Leibniz 6 discussed equations of the form x x z=24, s -f-2* = b y x x +z z = c. On May 9, 1694, Johann Bernoulli 7 mentions expressions of this sort in a letter to Leibniz who, in 1695, again considered exponentials in the Acta eruditorum, as did also Johann Bernoulli in 1697. 311. Of interest is the following quotation from a discussion by T. P. Nunn, in the Mathematical Gazette, Volume VI (1912), page 255, from which, however, it must not be inferred that Wallis actually wrote down fractional and negative exponents: "Those who are acquainted with the work of John Wallis will remember that he invented negative and fractional indices in the course of an investigation into methods of evaluating areas, etc. He had 1 David Gregory, Exercitatio geometrica de dimensione ftgurarum (Edinburgh, 1684), p. 4-6. 2 William Jones, Synopsis palmariorum matheseos (London, 1706), p. 67, 115-19. 3 Charles Reyneau, Analyse demontree (Paris, 1708), Vol. I, Introduction. 4 See G. Enestrom, BiUiotheca mathematica (2d ser.), Vol. XI (1897), p. 49. 8 P. H. Fuss, Correspondance mathematique et physique (Petersburg, 1843), Vol. I, p. 111. 6 C. I. Gerhardt, Brief wechsel von G. W. Leibniz mil Mathematikern (2d ed.; Berlin, 1899), Vol. I, p. 568. 7 Johann Bernoulli in Leibnizens Malhematische Schriften (ed. C. I. Gerhardt), Vol. Ill (1855), p. 140. POWERS 357 discovered that if the ordinates of a curve follow the law y~kx n its area follows the law -A= T-T ' kx n + l , n being (necessarily) a positive integer. This law is so remarkably simple and so powerful as a method that Wallis was prompted to inquire whether cases in which the ordi- k nates follow such laws as y y = k \/x could not be brought within x , its scope. He found that this extension of the law would be possible k ~ if could be written kx~", and k \/x as kx n . From this, from numerous *c other historical instances, and from general psychological observa- tions, I draw the conclusion that extensions of notation should be taught because arid when they are needed for the attainment of some practical purpose, and that logical criticism should come after the suggestion of an extension to assure us of its validity." 312. Notation for principal values. When in the early part of the nineteenth century the multiplicity of values of a n came to be studied, where a and n may be negative or complex numbers, and when the need of defining the principal values became more insistent, new nota- tions sprang into use in the exponential as well as the logarithmic theories. A. L. Cauchy 1 designated all the values that a n may take, for given values of a and n [a^O], by the symbol ((a))*, so that ((a))*^ e xia t 2kx*i f wncre i means the tabular logarithm of |a|, e = 2.718 . . . . , 7r = 3.141 . . . . , fc = 0, 1, 2, .... This notation is adopted by O. Stolz and J. A. Gmeiner 2 in their Theoretische Arithmetik. Other notations sprang up in the early part of the last century, Martin Ohm elaborated a general exponential theory as early as 1821 in a Latin thesis and later in his System der Mathematik ( 1822-33). 3 In a x , when a and x may both be complex, log a has an infinite num- ber of values. When, out of this infinite number some particular value of log a, say <x , is selected, he indicates this by writing (a|| ). With this understanding he can write x log a-\-y log a= (x-\-y) log a, and consequently a x <a y = a x+y is a complete equation, that is, an equa- tion in which both sides have the same number of values, representing exactly the same expressions. Ohm did not introduce the particular value of a x which is now called the "principal value." 1 A. L. Cauchy, Cours d' analyse (Paris, 1821), chap, vii, 1. 2 O. Stolz and J. A. Gmeiner, Theoretische Arithmetik (Leipzig), Vol. II (1902), p. 371-77. 3 Martin Ohm, Vcrsuch eines vollkommen consequcnten Systems der Mathe- matik, Vol. II (2d ed., 1829), p. 427. [First edition of Vol. II, 1823.] 358 A HISTORY OF MATHEMATICAL NOTATIONS Crelle 1 let \u\ k indicate some fixed value of u k , preferably a real value, if one exists, where k may be irrational or imaginary; the two vertical bars were used later by Weierstrass for the designation of absolute value ( 492). 313. Complicated exponents. When exponents themselves have exponents, and the latter exponents also have exponents of their own, then clumsy expressions occur, such as one finds in Johann I Ber- noulli, 2 Goldbach, 3 Nikolaus II Bernoulli, 4 and Waring. 2. Sit data exponentialis quantitas *' x v, & per przecedentem methodum inveniri poteft cjus fluxio **v 4* vytf~*x +. v x & x log. xxy. 3. Sit exponentialis quantitas & , & ejus fluxio erit y* x x* Ac, Arc, 4P * W ft v &c. w v w fte. * v t &C* x z -f- ^ xj>* x fc* x w* c x v v *' x log. ^ x log,^ x log. te * *e, v tp &e. ZXV-f-*" X^" X5J* XV^X &C. W**"-" 1 X log. ^ X log. y X. log. 2 x log. v x w 4- &c* unde facile conftabit lex, quam obfervat fcries* _ FIG. 106. E. Waring's ''repeated exponents." (From Meditationes analyticae [1785], p. 8.) De Morgan 5 suggested a new notation for cases where exponents are complicated expressions. Using a solidus, he proposes a A { (a-\-bx) /(c+ex)}j where the quantity within the braces is the exponent of a. He returned to this subject again in 1868 with the statement: "K convenient notation for repeated exponents is much wanted: not a working symbol, but a contrivance for preventing the symbol from wasting a line of text. The following would do perfectly well, #|a|&|c|d, 1 A. L. Crelle in Crelle's Journal, Vol. VII (1831), p. 265, 266. 2 lohann I Bernoulli, Ada eruditorum (1697), p. 125-33. P. II. Fuss, Correspondence math, et phys. ... du XVIII* siecle, Vol. II (1843), p. 128. 4 Op. cit., p. 133. 6 A. de Morgan, "Calculus of Functions," Encyclopaedia Metropolitana, Vol. II (1845), p. 388. POWERS 359 in which each post means all which follows is to be placed on the top of it. Thus: 1 x\a\b\c\d = x^ c \ d = x* blcld = ^ 6c|d = x^ ." When the base and the successive exponents are all alike, say a, Woepcke 2 used the symbol Q for((a a ) a "\ a and ^ for a (a"(^) where m indicates the number of repetitions of a. He extended this notation to cases where a is real or imaginary, not zero, and m is a positive or negative integer, or zero. A few years later J. W. L. Glaisher sug- gested still another notation for complicated exponents, namely, 1 T atxM -f, the arrows merely indicating that the quantity between x them is to be raised so as to become the exponent of a. Glaisher prefers this to "a Exp. u" for a u . Harkness and Morley 3 state, "It is usual to write exp (z)=e z , when z is complex." The contraction "exp" was recommended by a British Committee ( 725) in 1875, but was ignored in the suggestions of 1916, issued by the Council of the London Math- ematical Society. G. H. Bryan stresses the usefulness of this symbol. 4 Another notation was suggested by H. Schubert. If a is taken as an exponent of a, one obtains a^ or a a , and so on. Schubert desig- nates the result by (a; 6), indicating that a has been thus written 6 times. 5 For the expression (a;6) (tt;c) there has been adopted the sign (a; 6+c), so that (a; &)<<" c > = (a; c+l)<: *-. 314. D. F. Gregory 6 in 1837 made use of the sign (+) f , r an integer, to designate the repetition of the operation of multiplication. Also, (+a 2 ) i = + i (a 2 ) i =+*a> where the +* "will be different, according as we suppose the + to be equivalent to the operation repeated an even or an odd number of times. In the former case it will be equal to +, in the latter to . And generally, if we raise +a to any power m, whether whole or fractional, we have (+a) m = + m a m So long as 1 A. de Morgan, Transactions of the Cambridge Philosophical Society, Vol. XI, Part III (1869), p. 450. 2 F. Woepcke in Crelle's Journal, Vol. XLII (1851), p. 83. 3 J. Harkness and F. Morley, Theory of Functions (New York, 1893), p. 120. 4 Mathematical Gazette, Vol. VIII (London, 1917), p. 172, 220. 6 H. Schubert in Encyclopedic d. scien. math., Tome I, Vol. I (1904), p. 61. L. Euler considered aP y etc. See E. M. Le"meray, Proc. Edinb. Math. Soc., Vol. XVI (1897), p. 13. 8 The Mathematical Writings of Duncan Farquharson Gregory (ed. William Walton; Cambridge, 1865), p. 124-27, 145. 360 A HISTORY OF MATHEMATICAL NOTATIONS m is an integer, nn is an integer, and + rm a m has only one value; but if v r- m be a fraction of the form -, + will acquire different values, ac- cording as we assign different values to r .... i/( a)Xj/(~ a ) = l/(+a 2 ) ]/(-f )|/(a 2 ) = a; for in this case we know how the + has been derived, namely from the product = +, or 2 =+, which of course gives + * = , there being here nothing indeterminate about the +. It was in consequence of sometimes tacitly assuming the existence of +, and at another time neglecting it, that the errors in various trigonometrical expressions arose; and it was by the intro- duction of the factor cos 2r?H * sin 2rv (which is equivalent to + r ) that Poinsot established the formulae in a more correct and general P p shape." Gregory finds "sin (+ q c) = +<* sin c." A special notation for the positive integral powers of an imaginary root r of x n ~ l +x n ~ 2 + .... +x+l = 0, n being an odd prime, is given by Gauss; 1 to simplify the typesetting he designates r, rr, r 3 , etc., by the symbols [1], [2], [3], etc. 315. Conclusions. There is perhaps no symbolism in ordinary algebra which has been as well chosen and is as elastic as the Cartesian exponents. Descartes wrote a 3 , z 4 ; the extension of this to general exponents a" was easy. Moreover, the introduction of fractional and negative numbers, as exponents, was readily accomplished. The ir- rational exponent, as in a 1 / 2 , found unchallenged admission. It was natural to try exponents in the form of pure imaginary or of complex numbers (L. Euler, 1740). In the nineteenth century valuable inter- pretations were found which constitute the general theory of b n where b and n may both be complex. Our exponential notation has been an aid for the advancement of the science of algebra to a degree that could not have been possible under the old German or other early notations. Nowhere is the importance of a good notation for the rapid advancement of a mathematical science exhibited more forcibly than in the exponential symbolism of algebra. SIGNS FOR ROOTS 316. Early forms. Symbols for roots appear very early in the development of mathematics. The sign [p'for square root occurs in two Egyptian papyri, both found at Kahun. One was described by F. L. 1 C. F. Gauss, Disquisitions arithmeticae (Leipzig, 1801), Art. 342; Werke, Vol. I (1863), p. 420. ROOTS 361 Griffith 1 and the other by H. Schack-Schackenburg. 2 For Hindu signs see 107, 108, 112; for Arabic signs see 124. 317. General statement. The principal symbolisms for the desig- nation of roots, which have been developed since the influx of Arabic learning into Europe in the twelfth century, fall under four groups having for their basic symbols, respectively, R (radix), I (latus), the sign |/, and the fractional exponent. 318. The sign I}; first appearance. In a translation 8 from the Arabic into Latin of a commentary of the tenth book of the Elements of Euclid, the word radix is used for "square root." The sign R came to be used very extensively for "root/' but occasionally it stood also for 'the first power of the unknown quantity, x. The word radix was used for x in translations from Arabic into Latin by John of Seville and Gerard of Cremona ( 290). This double use of the sign R for x and also for square root is encountered in Leonardo of Pisa (122, 292) 4 and Luca Pacioli ( 135-37, 292). Before Pacioli, the use of R to designate square root is also met in a correspondence that the German astronomer Regiomontanus ( 126) 5 carried on with Giovanni Bianchini, who was court astronomer at Ferrara in Italy, and with Jacob von Speier, a court astronomer at Urbino ( 126). In German manuscripts referred to as the Dresden MSS C. 80, written about the year 1480, and known to have been in the hands of J. Widman, H. Grammateus, and Adam Riese, there is a sign con- sisting of a small letter, with a florescent stroke attached (Fig. 104). It has been interpreted by some writers as a letter r with an additional stroke. Certain it is that in Johann Widman's arithmetic of 1489 occurs the crossed capital letter R, and also the abbreviation ra (293). Before Widman, the Frenchman Chuquet had used R for "root" 1 F. L. Griffith, The Petrie Papyri, I. Kahun Papyri, Plate VIII. 2 H. Schack-Schackenburg in Zeitschrift fur aegyptische Sprache und Alter- tumskunde, Vol. XXXVIII (1900), p. 136; also Plate IV. See also Vol. XL, p. 65. 3 M. Curtzo, Anaritii in decent libros elementorum Euclidis commentarii (Leipzig, 1899), p. 252-386. 4 Scritti di Leonardo Pisano (ed. B. Boncompagni), Vol. II (Rome, 1862), "La practica geometriae," p. 209, 231. The word radix, meaning x, is found also in Vol. I, p. 407. 6 M. Curtze, "Der Briefwechsel Riomantan's, etc.," Abhandlungen zur Ge- schichte der mathematischen Wissenschaften, Vol. XII (Leipzig, 1902), p. 234, 318. 362 A HISTORY OF MATHEMATICAL NOTATIONS in his manuscript, Le Triparty ( 130). He 1 indicates ft* 16. as 4, "ft*. 16. si est . 2.," "# 5 . 32. si est . 2." 319. Sixteenth-century use of ft. The different uses of ft. made in Pacioli's Summa (1494, 1523) are fully set forth in 134-38. In France, De la Roche followed Chuquet in the use of ft. ( 132). The symbol appears again in Italy in Ghaligai's algebra (1521), and in later editions ( 139), while in Holland it appeared as early as 1537 in the arithmetic of Giel Van der Hoecke ( 150) in expressions like "Item wildi aftrecken ft | van ft | resi ft -J"; i.e., V\ V~\ = V\. The em- ployment of ft in the calculus of radicals by Cardan is set forth in 141, 199. A promiscuous adoption of different notations is found in the algebra of Johannes Scheubel ( 158, 159) of the University of Tubingen. He used Widman's abbreviation ra, also the sign j/i he indicates cube root by ra. cu. or by /vw/> fourth root by ra. ra. or by /W/. He suggests a notation of his own, of which he makes no further use, namely, radix se., for cube root, which is the abbreviation of radix secundae quantitatis. As the sum "ra. 15 ad ra. 17" he gives "ra. col. 32+1/1020," i.e., 1/15 +1/17 = v"32+ 1/1020. The col, collecti, signifies here aggregation. Nicolo Tartaglia in 1556 used ft extensively and also parentheses ( 142, 143). Francis Maurolycus 2 of Messina in 1575 wrote "r. 18" for 1/18, "r. v.Qm.r 7" for 1/6 1/7J. Bombelli's radical notation is explained in 144. It thus appears that in Italy the ft had no rival during the sixteenth century in the calculus of radicals. The only variation in the symbolism arose in the marking of the order of the radical and in the modes of designation of the aggregation of terms that were affected by ft. 320. In Spain 3 the work of Marco Aurel (1552) ( 204) employs the signs of Stifel, but Antich Rocha, adopting the Italian abbrevia- tions in adjustment to the Spanish language, lets, in his Arithmetica of 1564, "15 Mas ra. q. 50 Mas ra. q. 27 Mas ra. q. 6" stand for 15+1/50+ 1/27+1/6. A few years earlier, J. Perez de Moya, in his Aritmetica practica y speculativa (1562), indicates square root by r, 1 Le Triparty en la science des nombres par Maistre Nicolas Chuquet Parisien ... par M. Aristide Marre in Boncompagni's Bullettino, Vol. XIII, p. 655; (reprint, Rome, 1881), p. 103. 2 D. Francisci Mavrolyci, Abbatis Messanensi8 t Opuscula mathematica (Venice, 1575), p. 144. 3 Our information on these Spanish authors is drawn partly from Julio Rey Pastor, Los Matemdticos espanoles de siglo XVI (Oviedo, 1913), p. 42. ROOTS 363 cube root by rrr, fourth root by rr, marks powers by eo., ce., cu., c. ce., and "plus" by p, "minus" by ra, "equal" by eq. In Holland, Adrianus Romanus 1 used a small r, but instead of v wrote a dot to mark a root of a binomial or polynomial; he wrote r bin. 2+r bin. 2+r bin. 2+r 2. to designate In Tartaglia's arithmetic, as translated into French by Gosselin 2 of Caen, in 1613, one finds the familiar /? cu to mark cube root. A modification was introduced by the Scotsman James Hume, 3 residing in Paris, who in his algebra of 1635 introduced Roman numerals to indicate the order of the root ( 190). Two years later, the French text by Jacqves de Billy 4 used 5Q, &C, RQC for v 7 "", V ~~, i/~~> respectively. 321. Seventeenth-century use of R. During the seventeenth cen- tury, the symbol R lost ground steadily but at the close of the century it still survived; it was used, for instance, by Michael Rolle 5 who em- ployed the signs 2+R. 121. to represent 2-fV 7 121, and R. trin. 6oa66-9a 4 6-6 3 to represent i/6a 2 6 2 -- 9a 4 6 -- 6 8 ~. In 1690 H. Vitalis 6 takes Ri to represent secunda radix, i.e., the radix next after the square root. Consequently, with him, as with Scheubel, 3. R. 2* 8, meant 3f x/ 8, or 6. The sign R or #, representing a radical, had its strongest foothold in Italy and Spain, and its weakest in England. With the close of the seventeenth century it practically passed away as a radical sign; the symbol \/ gained general ascendancy. Elsewhere it will be pointed out in detail that some authors employed R to represent the unknown x. Perhaps its latest regular appearance as a radical sign is in the Spanish text of Perez de Moya ( 320), the first edition of which ap- peared in 1562. The fourteenth edition was issued in 1784; it still gave rrr as signifying cube root, and rr as fourth root. Moya's book offers a most striking example of the persistence for centuries of old and clumsy notations, even when far superior notations are in general use. 1 Ideae Mathematical Pars Prima, .... Adriano Romano Lovaniensi (Ant- werp, 1593), following the Preface. 2 U Arithmetique de Nicolas Tartaglia Brescian, traduit ... par Gvillavmo Gosselin de Caen, Premier Partie (Paris, 1613), p. 101. 3 James Hume, Traite de Valgebre (Paris, 1635), p. 53. 4 Jacqves de Billy, Abrege des Preceples d'Algebre (Rheims, 1637), p. 21. 6 Journal des Scavans de TAn 1683 (Amsterdam, 1709), p. 97. 6 Lexicon mathematicum ... authore Hieronymo Vitali (Rome, 1690), art. "Algebra." 364 A HISTORY OF MATHEMATICAL NOTATIONS 322. The sign 1. The Latin word latus ("side of a square") was introduced into mathematics to signify root by the Roman surveyor Junius Nipsus, 1 of the second century A.D., and was used in that sense by Martianus Capella, 2 Gerbert, 3 and by Plato of Tivoli in 1145, in his translation from the Arabic of the Liber embadorum ( 290). The symbol I (latus) to signify root was employed by Peter Ramus 4 with whom "I 27 ad I 12" gives "I 75," i.e., 1/27+1/12 = 1/76; "II 32 de II 162" gives "tt 2," i.e., 1/32 from I 7 162= ^2. Again, 5 "8-/ 20 in 2 quoins est (4-J 5." means 8-1/20, divided by 2, gives the quotient 4 j/5- Similarly, 6 "Ir. /1 12 /76" meant 1/1/112 1/76; the r signifying here residua, or "remainder," and therefore Ir. signified the square root of the binomial difference. In the 1592 edition 7 of Ramus' arithmetic and algebra, edited by Lazarus Schoner, "Ic 4" stands for (^4, and "I bq 5" for 1/5, in place of Ramus' "II 5." Also, 1/2. 1/3 = 1/6, 1/6 -v- 1/2 = 1/3 is expressed thus: 8 "Esto multiplicandum 1 2 per J 3 factus erit I 6. 12. I6f 13. /2V 3 ' 16. It is to be noted that with Schoner the I received an extension of meaning, so that 51 and Z5, respectively, represent 5x and 1/5, the I standing for the first power of the unknown quantity when it is not 1 Die Schriften der romischen Feldmesser (ed. Blume, Lachmann, Rudorff; Berlin, 1848-52), Vol. I, p. 96. 2 Martianus Capella, DeNuptiis (ed. Kopp; Frankfort, 1836), lib. VII, 748. 3 Gerberti opera mathematica (ed. Bubnow; Berlin, 1899), p. 83. See J. Tropfke, op. cit., Vol. II (2d ed., 1921), p. 143. 4 P. Rami Scholarvm mattiematicarvm libri unus et triginti (Basel, 1569), Lib. XXIV, p. 276, 277. 6 Ibid., p. 179. *Ibid., p. 283, 7 Petri Rami ... Arithmeticea libri duo, et algebrae totidem: a Lazaro Schoner o (Frankfurt, 1592), p. 272 ff. 8 Petri Rami ... Arithmetical libri duo et geometriae septem et viginti t Dudum quidem, a Lazaro Schonero (Frankfurt a/M., 1627), part entitled "De Nvmeri figvratis Lazari Schoneri liber/ ' p. 178. ROOTS 365 followed by a number (see also 290). A similar change in meaning resulting from reversing the order of two symbols has been observed in Pacioli in connection with 5 ( 136, 137) and in A. Girard in connection with the circle of Stevin ( 164). The double use of the sign I, as found in Schoner, is explained more fully by another pupil of Rarnus, namely, Bernardus Salignacus ( 291). Ramus' I was sometimes used by the great French algebraist Francis Victa who seemed disinclined to adopt either R or ]/ for indicating roots ( 177). This use of the letter I in the calculus of radicals never became popular. After the invention of logarithms, this letter was needed to mark logarithms. For that reason it is especially curious that Henry Briggs, who devoted the latter part of his life to the computation and the algorithm of logarithms, should have employed I in the sense as- signed it by llamus and Vieta. In 1624 Briggs used /, i( 3 ), II for square, cube, and fourth root, respectively. "Sic / (3 ) 8 [i.e., i ? 8], latus cubicum Octonarii, id est 2. sic / bin 2 -f 13. [i.e., ^ 2+1/3] latus binomii 2+Z3." Again, "II 85 J [i.e., ^8511. Latus 85 J est O**-* 4 - 30 -*, et huius lateris latus est S 03 -- 51 - 3 - 43 " 2 - 4 -. cui numero aequatur // 85J." 1 323. Napier 1 s line symbolism. John Napier 2 prepared a manu- script on algebra which was not printed until 1839. He made use of StifcFs notation for radicals, but at the same time devised a new scheme of his own. "It is interesting to notice that although Napier invented an excellent notation of his own for expressing roots, he did not make use of it in his algebra, but retained the cumbrous, and in some cases ambiguous notation generally used in his day. His nota- tion was derived from this figure 1 |2 4 | 5 71 8 3 9 in the following way: U prefixed to a number means its square root, ID its fourth root, D its fifth root, T its ninth root, and so on, with extensions of obvious kinds for higher roots. " 3 1 Henry Briggs, Arithmetica logarithmica (London, 1624), Introduction. 2 De Arte Logistica Joannis Naperi Merchistonii Baronis Libri qui super aunt (Edinburgh, 1839), p. 84. 3 J. E. A. Steggall, "De arte logistica," Napier Tercentenary Memorial Volume (ed. Cargill Gilston Knott; London, 1915), p. 160. 366 A HISTORY OF MATHEMATICAL NOTATIONS THE SIGN l/ 324. Origin of i/. This symbol originated in Germany. L. Euler guessed that it was a deformed letter r, the first letter in radix. 1 This opinion was held generally until recently. The more careful study of German manuscript algebras and the first printed algebras has con- vinced Germans that the old explanation is hardly tenable; they have accepted the a priori much less probable explanation of the evolution of the symbol from a dot. Four manuscript algebras have been avail- able for the study of this and other questions. The oldest of these is in the Dresden Library, in a volume of manu- scripts which contains different algebraic treatises in Latin and one in German. 2 In one of the Latin manuscripts (see Fig. 104, A 7), probably written about 1480, dots are used to signify root extraction. In one place it says: "In extraccione radicis quadrati alicuius numeri preponatur nurnero vnus punctus. In extraccione radicis quadrati radicis quadrati prepone numero duo puncta. In extraccione cubici radicis alicuius numeri prepone tria puncta. In extraccione cubici radicis alicuius radicis cubici prepone 4 puncta." 3 That is, one dot () placed before the radicand signifies square root; two dots (..) signify the square root of the square root; three dots (...) signify cube root; four dots (....), the cube root of the cube root or the ninth root. Evi- dently this notation is not a happy choice. If one dot meant square root and two dots meant square root of square root (i.e., 1/j/ ), then three dots should mean square root of square root of square root, or eighth root. But such was not actually the case; the three dots were made to mean cube root, and four dots the ninth root. What was the origin of this dot-system? No satisfactory explanation has been found. It is important to note that this Dresden manuscript was once in the possession of Joh. Widman, and that Adam Riese, who in 1524 prepared a manuscript algebra of his own, closely followed the Dresden algebra. 325. The second document is the Vienna MS 4 No. 5277, Ilegule- 1 L. Euler, Institutiones calculi differentialis (1775), p. 103, art. 119; J. Tropfke, 07?. cit., Vol. II (2d ed., 1921), p. 150. , 2 M. Cantor, Varies, uber Geschichte der Malhematik, Vol. II (2. Aufl., 1900), p. 241. 3 E. Wappler, Zur Geschichte der deulschen Algebra im XV. Jahrhundert, Zwickauer Gyrnnasialprogramm von 1887, p. 13. Quoted by J. Tropfke, op. tit., Vol. II (1921), p. 146, and by M. Cantor, op. tit., Vol. II (2. Aufl., 1900), p. 243. 4 C. J. Gerhardt, Monatsberichte Akad. (Berlin, 1867), p. 46; ibid. (1870), p. 143-47; Cantor, op. cit., Vol. II (2d ed., 1913), p. 240, 424. ROOTS 367 Cose-uel Algobre-. It contains the passage: "Quum 3 assimiletur radici de radice punctus deleatur de radice, 3 in se ducatur et remanet adhuc inter se aequalia"; that is, "When x 2 = V / x, erase the point before the x and multiply x 2 by itself, then things equal to each other are obtained." In another place one finds the statement, per punctum intellige radicem "by a point understand a root." But no dot is actually used in the manuscript for the designation of a root. The third manuscript is at the University of Gottingen, Codex Gotting. Philos. 30. It is a letter written in Latin by Initius Algebras, 1 probably before 1524. An elaboration of this manuscript was made in German by Andreas Alexander. 2 In it the radical sign is a heavy point with a stroke of the pen up and bending to the right, thus /. It is followed by a symbol indicating the index of the root; /$ indi- cates square root; /c e , cube root; /cc% the ninth root, etc. More- over, /cs|8+/22j stands for * 8+1/22, where cs (i.e., communis) signifies the root of the binomial which is designated as one quantity, by lines, vertical and horizontal. Such lines are found earlier in Chuquet ( 130). The $, indicating the square root of the binomial, is placed as a subscript after the binomial. Calling these two lines a "gnomon," M\ Curtze adds the following: "This gnomon has here the signification, that what it embraces is not a length, but a power. Thus, the simple 8 is a length or simple number, while [83 is a square consisting of eight areal units whose linear unit is /$|8. In the same way [8 C * would be a cube, made up of 8 cubical units, of which fc e 8 is its side, etc. A double point, with the tail attached to the last, signifies always the root of the root. For example, ,/c c [88 would mean the cube root of the cube root of 88. It is identical with /cc'88, but is used only when the radicand is a so-called median [Mediale] in the Euclidean sense." 3 326. The fourth manuscript is an algebra or Coss completed by Adam Riese 4 in 1524; it was not printed until 1892. Riese was familiar with the small Latin algebra in the Dresden collection, cited above; 1 Initius Algebras: Algebrae Arabis Arithmetici viri clarrisimi Liber ad Ylem geometram magislrum suum. This was published by M. Curtze in Abhandlungen zur Geschichte der mathematischen Wissenschaften, Heft XIII (1902), p. 435-611. Matters of notation are explained by Curtze in his introduction, p. 443-48. 2 G. Enestrom, Bibliotheca mathematica (3d ser.), Vol. Ill (1902), p. 355-60. 3 M. Curtze, op. cit. t p. 444, 4 B. Berlet, Adam Riese, sein Leben, seine Rechenbucher und seine Art zu rechnen; die Coss von Adam Riese (Leipzig-Frankfurt a/M., 1892). 368 A HISTORY OF MATHEMATICAL NOTATIONS he refers also to Andreas Alexander. 1 For indicating a root, Riese does not use the dot, pure and simple, but the dot with a stroke attached to it, though the word punct ("point ") occurs. Riese says: "1st, so $ vergleicht wird j/ vom radix, so mal den g in sich multipli- ciren vnnd das punct vor dem Radix aussleschn." 2 This passage has the same interpretation as the Latin passage which we quoted from the Vienna manuscript. We have now presented the main facts found in the four manu- scripts. They show conclusively that the dot was associated as a sym- bol with root extraction. In the first manuscript, the dot actually appears as a sign for roots. The dot does not appear as a sign in the second manuscript, but is mentioned in the text. In the third and fourth manuscripts, the dot, pure and simple, does not occur for the designation of roots; the symbol is described by recent writers as a dot with a stroke or tail attached to it. The question arises whether our algebraic sign j/ took its origin in the dot. Recent German writers favor that view, but the evidence is far from conclusive. Johannes Widman, the author of the Rechnung of 1489, was familiar with the first manuscript which we cited. Nevertheless he does not employ the dot to designate root, easy as the symbol is for the printer. He writes clown $ and ra. Christoff Rudolff was familiar with the Vienna manuscript which uses the dot with a tail. In his Coss of 1525 he speaks of the Punkt in connection with root symbolism, but uses a mark with a very short heavy downward stroke (almost a point), followed by a straight line or stroke, slanting upward (see Fig. 59). As late as 1551, Scheubel, 3 in his printed Algebra, speaks of points. He says: "Solent tamen multi, et bene etiam, has desideratas radices, suis punctis cu linea quadam a dextro latere ascendente, notare " ("Many are accustomed, and quite appropriately, to designate the desired roots by points, from the right side of which there ascends a kind of stroke/ 1 ) It is possible that this use of "point" was technical, signifying "sign for root," just as at a later period the expression "decimal point" was used even when the symbol actually written down to mark a decimal fraction was a comma. It should be added that if Rudolff looked upon his radical sign as really a dot, he would have been less likely to have used the dot again for a second purpose in his radical symbolism, namely, for the purpose of designating that 1 B. Berlet, op. tit., p. 29, 33. 2 C. I. Gerhardt, op. cit. (1870) p. 151. 3 J. Scheubel, Algelvra compendiosa (Paris, 1551), fol. 25B. Quoted from J. Tropfke, op. cil. t Vol. II (2d ed., 1921), p. 149. ROOTS 369 the root extraction must be applied to two or more terms following the I/; this use of the dot is shown in 148. It is possible, perhaps prob- able, that the symbol in Rudolff and in the third and fourth manu- scripts above referred to is not a point at all, but an r, the first letter in radix. That such was the understanding of the sixteenth-century Spanish writer, Perez de Moya ( 204), is evident from his designa- tions of the square root by r, the fourth root by rr, and the cube root by rrr. It is the notation found in the first manuscript which we cited, except that in Moya the r takes the place of the dot ; it is the notation of Rudolff, except that the sign in Rudolff is not a regularly shaped r. In this connection a remark of H. Wieleitner is pertinent: "The dot appears at times in manuscripts as an abbreviation for the syllable ra. Whether the dot used in the Dresden manuscript represents this normal abbreviation for radix does not appear to have been specially examined. J)l The history of our radical sign j/, after the time of Rudolff, relates mainly to the symbolisms for indicating (1) the index of the root, (2) the aggregation of terms when the root of a binomial or polynomial is required. It took over a century to reach some sort of agreement on these points. The signs of Christoff Rudolff are explained more fully in 148. Stifel's elaboration of the symbolism of Andreas Alex- ander as given in 1544 is found in 153, 155. Moreover, he gave to the y its modern form by making the heavy left-hand, downward stroke, longer than did Rudolff. 327. Spread of y. The German symbol of y for root found its way into France in 1551 through Scheubers publication (159); it found its way into Italy in 1608 through Clavius; it found its way into England through Recorde in 1557 ( 168) and Dee in 1570 ( 169); it found its way into Spain in 1552 through Marco Aurel ( 165, 204), but in later Spanish texts of that century it was super- seded by the Italian /?. The German sign met a check in the early works of Vieta who favored Ramus' Z, but in later editions of Vieta, brought out under the editorship of Fr. van Schooten, the sign y displaced Vieta's earlier notations ( 176, 177). In Denmark Chris. Dibuadius 2 in 1605 gives three designations of square root, y 7 , i/Q, j/g; also three designations of cube root, 1/C, j/c, yc e ] and three designations of the fourth root, j/j/, i/QO, 1/JJ. Stifel's mode of indicating the order of roots met with greater 1 H. Wieleitner, Die Sicben Rechnungsarten (Leipzig-Berlin, 1912), p. 49. 2 C. Dibvadii in Arithmeticam irrationalivm Euclidis (Arnhem, 1005). 370 A HISTORY OF MATHEMATICAL NOTATIONS general favor than Rudolff's older and clumsier designation ( 153, 155). 328. Rudolff's signs outside of Germany. The clumsy signs of Christoff Rudolff, in place of which Stifel had introduced in 1544 and 1553 better symbols of his own, found adoption in somewhat modified form among a few writers of later date. They occur in Aurel's Spanish Arithmetica t 1552 ( 165). They are given in Recorde, Whetstone of Witte (1557) ( 168), who, after introducing the first sign, j/., pro- ceeds: "The seconde signe is annexed with Surde Cubes, to expresse their rootes. As this ./w\/16 whiche signifieth the Cubike roote of .16. And ./VW/.20. betokeneth the Cubike roote of .20. Andsoforthe. But many tymes it hath the Cossike signe with it also: as /wv/-c* 25 the Cubike roote of .25. And /wv / -c*.32. the Cubike roote of .32. The thirde figure doeth represente a zenzizenzike roote. As ./\\/.l2. is the zenzizenzike roote of .12. And /vs/.35. is the zenzizenzike roote of .35. And likewaies if it haue with it the Cossike signe .gj. As A\/3324 the zenzizenzike roote of .24. and so of other." The Swiss Ardiiser in 1627 employed Rudolff's signs for square root and cube root. 1 J. H. Rahn in 1659 used /vv\/ for evolution, 2 which may be a modified symbol of Rudolff; Rahn's sign is adopted by Thomas Brancker in his English translation of Rahn in 1668, also by Edward Hatton 3 in 1721, and by John Kirkby 4 in 1725. Ozanam 5 in 1702 writes V // 5+ v'2 and also M/5+WA/2. Samuel Jeake 6 in 1696 gives modifications of Rudolff's signs, along with other signs, in an elaborate explanation of the "characters" of "Surdes"; j/ means root, j/: or V or VV universal root, AA/ or Vl square root, AW/ or ]/</> cube root, /vwx/ or j/jj squared square root, /wwv/ or j/p sursolide root. On the Continent, Johann Caramuel 7 in 1670 used j/ for square root and repeated the symbol i/j/ for cube root: "i/j/27. est Radix Cubica Numeri 27. hoc est, 3." 1 Johann Ardiiser, Geometriae Thearicae et Practicae, XII. Bucher (Zurich, 1627), fol. 8L4. 2 Johann Heinrich Rahn, Teuische Algebra (Zurich, 1659). 8 Edward Hatton, An Intire System of Arithmetic (London, 1721), p. 287. 4 John Kirkby, Arithmetical Institutions (London, 1735), p. 7. 6 J. Ozanam, Nouveaux Siemens d'algebre ... par M. Ozanam, I. Par tie (Am- sterdam, 1702), p. 82. 8 Samuel Jeake, AOHSTIKHAOrf A, or Arithmetick (London, 1696), p. 293. 1 Joannis Caramvdis Mathesis Biceps. Veiu8> et Nova (Campaniae, 1670), p. 132. ROOTS 371 329. Stevin's numeral root-indices. An innovation of considerable moment were Stevin's numeral indices which took the place of Stifel's letters to mark the orders of the roots. Beginning with Stifel the sign ]/ without any additional mark came to be interpreted as mean- ing specially square root. Stevin adopted this interpretation, but in the case of cube root he placed after the j/ the numeral 3 inclosed in a circle ( 162, 163). Similarly for roots of higher order. Stevin's use of numerals met with general but not universal adoption. Among those still indicating the order of a root by the use of letters was Des- cartes who in 1637 indicated cube root by j/C. But in a letter of 1640 he 1 used the 3 and, in fact, leaned toward one of Albert Gjrard's notations, when he wrote |/3).20+|/392 for ^20+1/392. But very great diversity prevailed for a century as to the exact position of the numeral relative to the |/. Stevin's j/, followed by numeral indices placed within circles, was adopted by Stampioen, 2 and by van Schooten. 3 A. Romanus displaced the circle of Stevin by two round parenthe- ses, a procedure explained in England by Richard Sault 4 who gives or a+b\*. Like Girard, Harriot writes j/3.)26+i/675 for ^26 +1/675 (see Fig. 87 in 188). Substantially this notation was used by Descartes in a letter to Mersenne (September 30, 1640), where he represents the ratine cubique by i/3), the ratine sursolide by 1/5), the B sursolide by j/7), and so on. 5 Oughtred sometimes used square brackets, thus v/[12]1000 for I^IOOO ( 183). 330. A step in the right direction is taken by John Wallis 6 who in 1655 expresses the root indices in numerals without inclosing them in a circle as did Stevin, or in parentheses as did Romanus. However, Wallis' placing of them is still different from the modern; he writes j/ 3 /2 2 for our f/R 2 . The placing of the index within the opening of the radical sign had been suggested by Albert Girard as early as 1629. Wallis' notation is found in the universal arithmetic of the Spaniard, Joseph Zaragoza, 7 who writes j/ 4 243 - y*%j for our 1^243- ^27, and 1 (Euvres de Descartes, Vol. X, p. 190. 2 Algebra ofte Nieuwe Stel-Regel ... door Johan Stampioen d'Jongle 's Graven- Hage (1639), p. 11. 3 Fr. van Schooten, Geometria a Renato des Cartes (1649), p. 328. 4 Richard Sault, A New Treatise of Algebra (London, n.d.). 6 (Euvres de Descartes, Vol. Ill (1899), p. 188. 6 John Wallis, Arilhmetica infinitorum (Oxford, 1655), p. 59, 87, 88. 7 Joseph Zaragoza, Arithmetica universal (Valencia, 1669), p. 307. 372 A HISTORY OF MATHEMATICAL NOTATIONS l/ 2 (7+i/ 2 13) for our ^7 +1/13. Wallis employs this notation 1 again in his Algebra of 1685. It was he who first used general indices 2 in the expression i/ d R d R. The notation i/ 4 19 for 1/19 crops out again 3 in 1697 in De Lagny's j/ 3 54 1/ 3 16 = j/ 3 2; it is employed by Thomas Walter; 4 it is found in the Maandelykse Mathematische Lief- hebberye (1754-69), though the modern \/ is more frequent; it is given in Castillion's edition 5 of Newton's Arithmetica universalis. 331. The Girard plan of placing the index in the opening of the radical appears in M. llolle's Traite d'Algebre (Paris, 1690), in a letter of Leibniz 6 to Varignon of the year 1702, in the expression 1/1 + j/ 3, and in 1708 in (a review of) G. Manfred 7 with literal index, *\aa+bb n . At this time the Leibnizian preference for i/(aa+bb) in place of l/aa+66 is made public; 8 a preference which was heeded in Germany and Switzerland more than in England and France. In Sir Isaac Newton's Arithmetica universalis 9 of 1707 (written by Newton some- time between 1673 and 1683, and published by Whiston without hav- ing secured the consent of Newton) the index numeral is placed after the radical, and low, as in |/3 : 64 for 1/64, so that the danger of con- fusion was greater than in most other notations. During the eighteenth century the placing of the root index in the opening of the radical sign gradually came in vogue. In 1732 one finds 1/25 in De la Loubere; 10 De Lagny 11 who in 1697 wrote j/ 3 , in 1733 wrote ^/~\ Christian Wolff 12 in 1716 uses in one place the astro- 1 John Wallis, A Treatise of Algebra (London, 1685), p. 107; Opera, Vol. II (1693), p. 118. But see also Arithmetica infmitorum (1656), Prop. 74. 2 Mathesis universalis (1657), p. 292. 3 T. F. de Lagny, Nouveaux elemens d'arithmetique el d'algcbre (Paris, 1697), p. 333. 4 Thomas Walter, A new Mathematical Dictionary (London, n.d., but pub- lished in 1762 or soon after), art. "Heterogeneous Surds." 5 Arithmetica universalis .... auclore Is. Newton .... cum commenlario Johannis Castillionei . . . . , Tomus primus (Amsterdam, 1761), p. 76. Journal des S^avans, anne"e 1702 (Amsterdam, 1703), p. 300. 7 Ibid., ann6e 1708, p. 271. 8 Ibid. 9 Isaac Newton, Arithmetica universalis (London, 1707), p. 9; Tropfke, Vol. II, p. 154. 10 Simon de la Loubere, De la Resolution des Equations (Paris, 1732), p. 119. 11 De Lagny in Memoir es de Vacademie r, des sciences, Tome XI (Paris, 1733), p. 4. 12 Christian Wolff, Mathematisches Lexicon (Leipzig, 1716), p. 1081. ROOTS 373 nomical character representing Aries or the ram, for the radical sign, and writes the index of the root to the right; thus T 3 signifies cube root. Edward Hatton 1 in 1721 uses f, \/, ty\ De la Chapelle 2 in 1750 wrote tyW. Wolff 3 in 1716 and Hindenburg 4 in 1779 placed the index to the left of the radical sign, 3 ]/Z; nevertheless, the notation \/ came to be adopted almost universally during the eighteenth century. Variations appear here and there. According to W. J, Greenstrcet, 6 a curious use of the radical sign is to be found in Walkingame's Tutor's Assistant (20th ed., 1784). He employs the letter V for square root, but lets F 3 signify cube or third power, F 4 the fourth power. On the use of capital letters for mathematical signs, very often encountered in old books, as F, for j/, > for >, Greenstreet remarks that "authors in the eighteenth century complained of the meanness of the Cam- bridge University Press for using daggers set sideways instead of the usual + " In 1811, an anonymous arithmetician 6 of Massachusetts suggests 2 4 for 1 / / 4, 3 8 for ^8, m 8 for l/'S. As late as 1847 one finds 7 the notaton 3 |/fe, m l/a&c, for the cube root and the wth root, the index appearing in front of the radical sign. This form was not adopted on account of the limitations of the print- ing office, for in an article in the same series, from the pen of De Morgan, the index is placed inside the opening of the radical sign. 8 In fact, the latter notation occurs also toward the end of Parker's book (p. 131). In a new algorithm in logarithmic theory A. Biirja 9 proposed the sign i/ a to mark the nth root of the order N, of a, or the number of which the nth power of the order N is a. 1 Edward Hatton, op. cit. (London, 1721), p. 287. 2 De la Chapelle, Traite des sections coniques (Paris, 1750), p. 15. 3 Christian Wolff, Mathematisches Lexicon (Leipzig, 1716), "Signa," p. 1265. 4 Carl F. Hindenburg, Infmitinomii dignitatum leges ac formulae (Gottingcn, 1779), p. 41. 5 W. J. Greenstreet in Mathematical Gazette, Vol. XI (1823), p. 315. 6 The Columbian Arithmetician, "by an American" (Havershall, Mass., 1811), p. 13. 7 Parker, "Arithmetic and Algebra," Library of Useful Knowledge (London, 1847), p. 57. 8 A. de Morgan, "Study and Difficulties of Mathematics," ibid., Mathematics, Vol. I (London, 1847), p. 56. 9 A. Biirja in Nouveaux memoires d. I' academic r. d. scienc. et bett.-lett. t anne"e 1778 et 1779 (Berlin, 1793), p. 322. 374 A HISTORY OF MATHEMATICAL NOTATIONS 332. Rudolff and Stifel's aggregation signs. Their dot symbolism for the aggregation of terms following the radical sign ]/ was used bv Peletier in 1554 ( 172). In Denmark, Chris. Dibuadius 1 in 1605 marks aggregation by one dot or two dots, as the case may demand. Thus v/.S+j/S. +1/2 means 1/5+1/3+1/2; i/.5+|/3+i/2means 1/3 + 1/2 . W. Snell's translation 2 into Latin of Ludolf van Ceulen's book on the circle contains the expression which is certainly neater than the modern V 2 The Swiss, Johann Ardtiser, 3 in 1627, represents i/(2 j/3) by V.2-*V3" and i/[2+i/(2+l/2)] by V-2+i/-2+i/2." This notation appears also in one of the manuscripts of Reri6 Descartes, 4 written before the publication of his Geomitrie in 1637. It is well known that Oughtred in England modified the German dot symbolism by introducing the colon in its place ( 181). He had settled upon the dot for the expression of ratio, hence was driven to alter the German notation for aggregation. Oughtred's 5 double colons appear as in "i/q:aq eq:" for our j/(a 2 e 2 ). We have noticed the use of the colon to express aggregation, in the manner of Oughtred, in the Arithmetique made easie, by Edmund Wingate (2d ed. by John Kersey; London, 1650), page 387; in John Wallis' Operum mathematicorum pars altera (Oxonii, 1656), page 186, as well as in the various parts of Wallis' Treatise of Algebra (London, 1685) ( 196), and also in Jonas Moore's Arithmetick in two Books (London, 1660), Second Part, page 14. The 1630 edition of Wingate's book does not contain the part on algebra, nor the symbolism in question; these were probably added by John Kersey. 1 C. Dibvadii in arithmeticam irrationalivm Evdidis (Arnhem, 1605), Intro- duction. 2 Lvdolphi d Cevlen de Circvlo Adscriptis Liber ... omnia 6 vernaculo Latina fecit ... Willebrordus Sndlius (Ley den, 1610), p. 5. 3 Johann Ardliser, Geometriae, Theweticae practicae, XII. Biicher (Zurich, 1627), p. 97, 98. 4 (Euvres de Descartes, Vol. X (1908) p. 248. 8 Eudidis dedaratio, p. 9, in Oughtred's Ctavis (1652). ROOTS 375 333. Descartes 9 union of radical sign and vinculum. Ren6 Des- cartes, in his Geometric (1637), indicates the cube root by j/C. as in for our Here a noteworthy innovation is the union of the radical sign }/ with the vinculum ( 191). This union was adopted in 1640 by J. J. Stampioen, 1 but only as a redundant symbol. It is found in Fr. van Schooten's 1646 edition of the collected works of Vieta ( 177), in van Schooten's conic sections, 2 as also in van Schooten's Latin edi- tion of Descartes' geometry. 8 It occurs in J. H. Rahn's algebra (1659) and in Brancher's translation of 1668 ( 194). This combination of radical sign -[/ and vinculum is one which has met with great favor and has maintained a conspicuous place in mathematical books down to our own time. Before 1637, this combi- nation of radical sign and vinculum had been suggested by Descartes (CEuvres, Vol. X, p. 292). Descartes also leaned once toward Girard's notation. Great as were Descartes' services toward perfecting algebraic notation, he missed a splendid opportunity of rendering a still greater service. Before him Oresme and Stevin had advanced the concept of fractional as well as of integral exponents. If Descartes, instead of extending the application of the radical sign j/ by adding to it the vinculum, had discarded the radical sign altogether and had intro- duced the notation for fractional as well as integral exponents, then it is conceivable that the further use of radical signs would have been discouraged and checked; it is conceivable that the unnecessary dupli- cation in notation, as illustrated by 6* and 1/6 3 , would have been avoided; it is conceivable that generations upon generations of pupils would have been saved the necessity of mastering the operations with two difficult notations when one alone (the exponential) would have answered all purposes. But Descartes missed this opportunity, as did later also I. Newton who introduced the notation of the fractional exponent, yet retained and used radicals. 1 J. J. Stampioen, Wis-Konstich ende Reden-Maetich Bewijs (The Hague, 1640), p. 6. 2 Francisd & Schooten Leydensis de organica conicarum sectionum (Leyden, 1646), p. 91. 3 Francisci a Schooten, Renati Descartes Geomelria (Frankfurt a/M., 1695), p. 3. [First edition, 1649.] 376 A HISTORY OF MATHEMATICAL NOTATIONS 334. Other signs of aggregation of terms. Leonard and Thomas Digges, 1 in a work of 1571, state that if "the side of the Pentagon, [is] 14, the containing circles scmidiameter [is] V / S.F.98+i/sl920f ' i.e., VQS+ i/1920| . In the edition of 159 1 2 the area of such a pentagon is given as 1/3 mi. 60025+1/32882400500 . Vieta's peculiar notations for radicals of 1593 and 1595 are given in 177. The Algebra of Herman Follinvs 3 of 1622 uses parentheses in connection with the radical sign, as in V / g(22+V / g9), our 1/22 +1/9. Similarly, Albert Girard 4 writes i/(2J + l/3-J), with the simplifica- tion of omitting in case of square root the letter marking the order of the root. But, as already noted, he does not confine himself to this notation. In one place 5 he suggests the modern designation i/ 7 , f/, i/. Oughtred writes i/u or j/b for universal root ( 183), but more commonly follows the colon notation ( 181). Herigone's notation of 1634 and 1644 is given in 189. The Scotsman, James Gregory, 6 writes J 5 1fi2 2 \ In William Molyneux 7 one finds VCP*-Px* for Vpi-Px*. Another mode of marking the root of a binomial is seen in a paper of James Bernoulli 8 who writes >/, ax x 2 for Vax x 2 . This is really the old idea of Stifel, with Herigone's and Leibniz's comma taking the place of a dot. The union of the radical sign and vinculum has maintained itself widely, even though it had been discouraged by Leibniz and others who aimed to simplify the printing by using, as far as possible, one- line symbols. In 1915 the Council of the London Mathematical So- 1 A Geometrical Practise, named Pantometria, framed by Leonard Digges t .... finished by Thomas Digges his sonne (London, 1571) (pages unnumbered). 2 A Geometrical Practical Treatize named Pantometria (London, 1591), p. 106. 3 Hermann Follinvs, Algebra sive liber de Rebvs occvltis (Cologne, 1622), p. 157. 4 Albert Girard, Invention nouvelle en l j algebre (Amsterdam, 1629). 5 Loc. cit.y in "Caracteres de puissances et racines." 6 James Gregory, Geometriae pars vniversalis (Patavii, 1668), p. 71, 108. 7 William Molyneux, A Treatise of Dioptricks (London, 1692), p. 299. 8 Jacob Bernoulli in Ada eruditorum (1697), p. 209. ROOTS 377 ciety, in its Suggestions for Notation and Printing , l recommended that 1/2 or 2* be adopted in place of 1/2, also i/(ax 2 +2bx+c) or (ax 2 + in place of V / ax 2 +2bx+c. Bryan 2 would write ]/ 1 rather than 1/^T. 335. Redundancy in the use of aggregation signs. J. J. Stampioen marked aggregation of terms in three ways, any one of which would have been sufficient. Thus, 3 he indicates l/6 3 +6a 2 6 2 +9 4 6 in this manner, l/.(666+6aa 66+9aaaa 6); he used here the vinculum, the round parenthesis, and the dot to designate the aggregation of the three terms. In other places, he restricts himself to the use of dots, either a dot at the beginning and a dot at the end of the expression, or a dot at the beginning and a comma at the end, or he uses a dot and parentheses. Another curious notation, indicating fright lest the aggregation of terms be overlooked by the reader, is found in John Kersey's symbolism of 1673, 4 1 /(2):Jr-l/irr-a: for \ r -V\r*-&. We observe here the superposition of two notations for aggregation, the Oughtredian colon placed before and after the binomial, and the vinculum. Either of these without the other would have been suffi- cient. 336. Peculiar Dutch symbolism. A curious use of |/ sprang up in Holland in the latter part of the seventeenth century and maintained itself there in a group of writers until the latter part of the eighteenth century. If ;/ is placed before a number it means "square root/' if placed after it means "square." Thus, Abraham de Graaf 5 in 1694 indicates by \/-^r ^ e square root of the fraction, by n~\ / the square \ 0*2 JJ5 \ of the fraction. This notation is used often in the mathematical jour- nal, Maandelykse Mathematische Liefhebberye, published at Amster- dam from 1754 to 1769. As late as 1777 it is given by L. Praalder 8 of Utrecht, and even later (1783) by Pieter Venema. 7 We have here the 1 Mathematical Gazette, Vol. VIII (1917), p. 172. 2 Op. cit., Vol. VIII, p. 220. 3 J. J. Stampioenii Wis-Konstigh Ende Reden-Maetigh Bewjs (The Hague, 1640), p. 6. 4 John Kersey, Algebra (London, 1673), p. 95. 6 Abraham de Graaf, De Geheele Mathesis (Amsterdam, 1694), p. 65, 69. 6 Laurens Praalder, Mathematische Voorstellen (Amsterdam, 1777), p. 14, 15 ff . 7 Pieter Venema, Algebra ofte Stel-Konst, Vyfde Druk (Amsterdam, 1783), p. 168, 173. 378 A HISTORY OF MATHEMATICAL NOTATIONS same general idea that was introduced into other symbolisms, accord- ing to which the significance of the symbol depends upon its relative position to the number or algebraic expression affected. Thus with Pacioli 5200 meant 1/200, but #3 meant the second power ( 135, 136). With Stevin ( 162, 163), 20 meant 20 3 , but 200 meant 20s 3 . With L. Schoner 5J meant 5z, but 15 meant 1/5 ( 291). We may add that in the 1730 edition of Venema's algebra brought out in New York City radical expressions do not occur, as I am informed by Professor L. G. Simons, but a letter placed on the left of an equation means division of the members of the equation by it; when placed on the right, multiplication is meant. Thus (p. 100) : 6 and (p. 112): " 5 _4500 . 1000 x x^ 5:^500 = Hz -1000 Similar is PrandePs use of -j/ as a marginal symbol, indicating that the square root of both sides of an equation is to be taken. His marginal symbols are shown in the following: 1 I/ 337. Principal root-values. For the purpose of distinguishing be- tween the principal value of a radical expression and the other values, G. Peano 2 indicated by i/*a all the m values of the radical, reserving i/a for the designation of its "principal value." This notation is adopted by O. Stolz and J. A. Gmeiner 3 in their Theoretische Arith- metik (see also 312). 1 J. G. Prandel, Kugldreyeckslehre und hohere Mathematik (Miinchen, 1793), p. 97. 2 G. Peano, Formidaire des math&natiques (first published in Rivista di Mate- matica ), Vol. I, p. 19. 3 0. Stolz und J. A. Gmeiner, Thevrelische Arithmetik (Leipzig), Vol. II (1902), p. 355. UNKNOWN NUMBERS 379 338. Recommendation of United States National Committee. "With respect to the root sign, i/, the committee recognizes that convenience of writing assures its continued use in many cases instead of the fractional exponent. It is recommended, however, that in algebraic work involving complicated cases the fractional exponent be preferred. Attention is called to the fact that the symbol I/a (a representing a positive number) means only the positive square root and that the symbol i^a means only the principal nth root, and i^ similarly for a*, a 71 ." 1 SIGNS FOR UNKNOWN NUMBERS 339. Early forms. Much has already been said on symbolisms used to represent numbers that are initially unknown in a problem, and which the algebraist endeavors to ascertain. In the Ahmes papyrus there are signs to indicate "heap" (23); in Diophantus a Greek letter with an accent appears ( 101); the Chinese had a positional mode of indicating one or more unknowns; in the Hindu Bakhshali manuscript the use of a dot is invoked ( 109). Brahmagupta and Bhaskara did not confine the symbolism for the unknown to a single sign, but used the names of colors to designate different unknowns ( 106, 108, 112, 114). The Arab Abu Kamil 2 (about 900 A.D.), modi- fying the Hindu practice of using the names of colors, designated the unknowns by different coins, while later al-Karkhi (following perhaps Greek sources) 3 called one unknown "thing," a second "measure" or "part," but had no contracted sign for them. Later still al-Qalasddi used a sign for unknown ( 124). An early European sign is found in Regiomontanus ( 126), later European signs occur in Pacioli ( 134, 136), in Christoff Rudolff ( 148, 149, 151), 4 in Michael Stifel who used more than one notation ( 151, 152), in Simon Stevin ( 162), in L. Schoner ( 322), in F. Vieta ( 176-78), and in other writers ( 117, 138, 140, 148, 164, 173, 175, 176, 190, 198). Luca Pacioli remarks 5 that the older textbooks usually speak of 1 Report of the National Committee on Mathematical Requirements under the Ausjyices of the Mathematical Association of America (1923), p. 81. 2 H. Suter, Bibliotheca mathematics (3d ser.), Vol. XI (1910-11), p. 100 ff. 3 F. Woepcke, Extrait du Fakhri (Paris, 1853), p. 3, 12, 13&-43. See M. Cantor, op. cit., Vol. I (3d cd., 1907), p. 773. 4 Q. Enestrom, Bibliotheca mathematics (3d ser.), Vol. VIII (1907-8), p. 207. 5 L. Pacioli, Summa, dist. VIII, tract 6, fol. 148#. See M. Cantor, op. cit., Vol. II (2d ed., 1913), p. 322. 380 A HISTORY OF MATHEMATICAL NOTATIONS the first and the second cosa for the unknowns, that the newer writers prefer cosa for the unknown, and quantita for the others. Pacioli abbreviates those co. and #K Vieta's convention of letting vowels stand for unknowns and consonants for knowns ( 164, 176) was favored by Albert Girard, and also by W. Oughtred in parts of his Algebra, but not throughout. Near the beginning Oughtred used Q for the unknown (182). The use of N (numerus) for x in the treatment of numerical equa- tions, and of Q, C, etc., for the second and third powers of x, is found in Xylander's edition of Diophantus of 1575 ( 101), in Vieta's De emen- datione aequationum of 1615 ( 178), in Bachet's edition of Diophantus of 1621, in Camillo Glorioso in 1627 ( 196). In numerical equations Oughtred uses / for x, but the small letters q, c, qq, gc, etc., for the higher powers of x ( 181). Sometimes Oughtred employs also the corresponding capital letters. Descartes very often used, in his corre- spondence, notations different from his own, as perhaps more familiar to his correspondents than his own. Thus, as late as 1640, in a letter to Mersenne (September 30, 1640), Descartes 1 writes "1C-6N = 40," which means x z 6# = 40. In the Regulae ad directionem ingenii, Descartes represents 2 by a, b, c, etc., known magnitudes and by A, J5, C, etc., the unknowns; this is the exact opposite of the use of these letters found later in Rahn. Crossed numerals representing powers of unknowns. Interest- ing is the attitude of P. A. Cataldi of Bologna, who deplored the existence of many different notations in different countries for the un- known numbers and their powers, and the inconveniences resulting from such diversity. He points out also the difficulty of finding in the ordinary printing establishment the proper type for the representation of the different powers. He proposes 3 to remove both inconveniences by the use of numerals indicating the powers of the unknown and dis- tinguishing them from ordinary numbers by crossing them out, so that 0, *, 2, 3, . , would stand for x, x', x 2 , x 3 Such crossed numerals, he argued, were convenient and would be found in printing offices since they are used in arithmetics giving the scratch method of dividing, called by the Italians the "a Galea" method. The reader will recall that Cataldi's notation closely resembles that of Leonard 1 CEuvres de Descartes, Vol. Ill (1899), p. 190, 196, 197; also Vol. XII, p. 279. 2 Op. cit. y Vol. X (1908), p. 455, 462. 3 P. A. Cataldi, Trattato dell' algebra proportionate (Bologna, 1610), and in his later works. See G. Wertheim in Bibliotheca mathemalica (3d ser.), Vol. II (1901), p. 146, 147. UNKNOWN NUMBERS 381 and Thomas Digges in England ( 170). These symbols failed of adoption by other mathematicians. We have seen that in 1627 Camillo Glorioso, in a work published at Naples/ wrote N for x, and q, c, qq, qc, cc, qqc, qcc, and ccc for x 2 , x 3 , . . . . , z 9 , respectively ( 196). In 1613 Glorioso had followed Stevin in representing an un- known quantity by 1O. 340. Descartes' z, y, x. The use of z, y, x . . . . to represent un- knowns is due to Rcn6 Descartes, in his La geometric (1637). Without comment, he introduces the use of the first letters of the alphabet to signify known quantities and the use of the last letters to signify unknown quantities. His own language is: "... 1'autre, LN, est \a la moitie de 1'autre quantity connue, qui estoit multipliee par z, quo ie suppose estre la ligne inconnue." 2 Again: "... ie considere ... Quo le segment de la ligne AB, qui est entre les poins A et B, soit nomine x, et que BC soit nomme' y; ... la proportion qui est entre les cost^s AB et BR est aussy donntfc, et ie la pose cornme de z a 6; de fagon qu' A B estant x, RB sera , et la toute CR sera y-\ . ..." Later he says: z z "et pour ce que CB et BA sont deux quantity's indetcrmme*es et in- connues, ie les nomme, Tune y; et 1'autre x. Mais, affin de trouver le rapport de Tune a 1'autre, ie considere aussy les quantites connues qui deterrninent la description de cete ligne courbe: comrne GA que jo nomine a, KL que je nomme b, et NL, parallele a GA, que ie nomine GV' 3 As co-ordinates he uses later only x and y. In equations, in the third book of the Geometric, x predominates. In manuscripts written in the interval 1629-40, the unknown z occurs only once. 4 In the other places x and y occur. In a paper on Cartesian ovals, 5 prepared before 1629, x alone occurs as unknown, y being used as a parameter. This is the earliest place in which Descartes used one of the last letters of the alphabet to represent an unknown. A little later he used x, y, z again as known quantities. 6 Some historical writers have focused their attention upon the x, disregarding the y and z, and the other changes in notation made by 1 Camillo Gloriosi, Exercitationes mathematical, decas I (Naples, 1627). Also Ad theorema geometricvm, d nobilissimo viro proposition, Joannis Camilli Gloriosi (Venice, 1613), p. 26. It is of interest that Glorioso succeeded Galileo in the mathe- matical chair at Padua. 2 (Euvres de Descartes, Vol. VI (1902), p. 375. 3 Ibid., p. 394. 4 Ibid., Vol. X, p. 288-324. 5 Ibid., p. 310. 6 Ibid., p. 299. 382 A HISTORY OF MATHEMATICAL NOTATIONS Descartes; these writers have endeavored to connect this x with older symbols or with Arabic words. Thus, J. Tropfke, 1 P. Treutlein, 2 and M. Curtze 8 advanced the view that the symbol for the unknown used by early German writers, 2, looked so much like an x that it could easily have been taken as such, and that Descartes actually did inter- pret and use it as an x. But Descartes' mode of introducing the knowns a, b, c, etc., and the unknowns z, T/, x makes this hypothesis improbable. Moreover, G. Enestrom has shown 4 that in a letter of March 26, 1619, addressed to Isaac Beeckman, Descartes used the symbol 3 as a symbol in form distinct from x, hence later could not have mistaken it for an 3 At one time, before 1637, Descartes 5 used x along the side of 3; at that time x, y, z are still used by him as symbols for known quantities. German symbols, including the 2 for x, as they are found in the algebra of Clavius, occur regularly in a manuscript 6 due to Descartes, the Opuscules de 1619-1621. All these facts caused Tropfke in 1921 to abandon his old view 7 on the origin of x, but he now argues with force that the resemblance of x and 2, and Descartes' familiarity with 2, may account for the fact that in the latter part of Descartes' Geometric the x occurs more frequently than z and y. Enestrom, on the other hand, inclines to the view that the predominance of x over y and z is due to typo- graphical reasons, type for x being more plentiful because of the more frequent occurrence of the letter x, to y and 2, in the French and Latin languages. 8 There is nothing to support the hypothesis on the origin of x due to Wertheim, 9 namely, that the- Cartesian x is simply the nota- tion of the Italian Cataldi who represented the first power of the unknown by a crossed "one," thus Z. Nor is there historical evidence 1 J. Tropfke, Geschichte der Elementar-Mathematik, Vol. I (Leipzig, 1902), p. 150. 2 P. Treutlein, "Die deutsche Coss," Abhandl. z. Geschichte d. mathematischen Wins., Vol. II (1879), p. 32. 3 M. Curtze, ibid., Vol. XIII (1902), p. 473. 4 G. Enestrom, Bibliotheca mathematica (3d ser.), Vol. VI (1905), p. 316, 317, 405, 406. See also his remarks in ibid. (1884) (Sp. 43); ibid. (1889), p. 91. ,Jhe letter to Beeckman is reproduced in (Euvres de Descartes, Vol. X (1908), p. 155. 6 (Euvres de Descartes, Vol. X (Paris, 1908), p. 299. See also Vol. Ill, Appendix II, No. 480. 6 Ibid., Vol. X (1908), p. 234. 7 J. Tropfke, op. cit., Vol. II (2d ed., 1921), p. 44-46. 8 G. Enestrom, Bibliotheca mathematica (3d ser.), Vol. VI, p. 317. 9 G. Enestrom, ibid. UNKNOWN NUMBERS 383 to support the statement found in Noah Webster's Dictionary, under the letter x, to the effect that "x was used as an abbreviation of Ar. shei a thing, something, which, in the Middle Ages, was used to desig- nate the unknown, and was then prevailingly transcribed as xei." 341. Spread of Descartes' signs. Descartes' x, y, and z notation did not meet with immediate adoption. J. II. Rahn, for example, says in his Teutsche Algebra (1659): "Descartes' way is to signify known quantities by the former letters of the alphabet, and unknown by the latter [z, y, x> etc.]. But I choose to signify the unknown quan- tities by small letters and the known by capitals." Accordingly, in a number of his geometrical problems, Rahn uses a and A, etc., but in the book as a whole he uses z, y, x freely. As late as 1670 the learned bishop, Johann Caramuel, in his Mathe- sis biceps ... , Campagna (near Naples), page 123, gives an old nota- tion. He states an old problem and gives the solution of it as found in Geysius; it illustrates the rhetorical exposition found in some books as late as the time of Wallis, Newton, and Leibniz. We quote: "Dicebat Augias Herculi: Meorum armentorurn media pars est in tali loco octavi in tali, decirna in tali, 20 a in tali 60 a in tali, & 50 . sunc hie. Et Geysius libr. 3 Cossa Cap. 4. haec pecora numeraturus sic scribit. "Finge 1. a. partes |a, |a, -^a, ^V a > V a & additae (hoc est, in summam reductae) sunt |fa & quibus de 1. a. sublatis, restant <& a aequalia 50. Jam, quia sictus, est fractio, multiplicando reducatur, & 1. a. aequantur 240. Hie est numerus pecorum Augiae." ("Augias said to Hercules: 'Half of my cattle is in such a place, | in such, t V m such, -sV m such, uV in such, and here there are 50. And Geysius in Book 3, Cossa, Chap. 4, finds the number of the herd thus: Assume La., the parts are Ja, ^a, -^a, ^a, -^a, and these added [i.e., reduced to a sum] are ||a which subtracted from 1. a, leaves ^ 4 a, equal to 50. Now, the fraction is removed by multiplication, and 1. a equal 240. This is the number of Augias' herd/ ") Descartes' notation, x, y, z, is adopted by Gerard Kinckhuysen, 1 in his Algebra (1661). The earliest systematic use of three co-ordinates in analytical geometry is found in De la Hire, who in his Nouveaux Siemens des sections coniques (Paris, 1679) employed (p. 27) x, y, v. A. Parent 2 used x, y, z; Euler 3 , in 1728, t y x, y; Joh. Bernoulli, 4 in 1715, 1 Gerard Kinckhuysen, Algebra ofte Slel-Konst (Haerlcm, 1661), p. 6. 2 A. Parent, Essais et recherches de math, et de phys., Vol. I (Paris, 1705). 3 Euler in Comm. Aca. Petr., II, 2, p. 48 (year 1728, printed 1732). 4 Leibniz and Bernoulli, Commerdum philosophicum et mathematicum, Vol. II (1745), p. 345. 384 A HISTORY OF MATHEMATICAL NOTATIONS x, y y z in a letter (February 6, 1715) to Leibniz. H. Pitot 1 applied the three co-ordinates to the helix in 1724. SIGNS OF AGGREGATION 342. Introduction. In a rhetorical or syncopated algebra, the aggregation of terms could be indicated in words. Hence the need for symbols of aggregation was not urgent. Not until the fifteenth and sixteenth centuries did the convenience and need for such signs definitely present itself. Various devices were invoked: (1) the hori- zontal bar, placed below or above the expression affected; (2) the use of abbreviations of words signifying aggregation, as for instance u or v for universalis or vniversalis, which, however, did not always indi- cate clearly the exact range of terms affected ; (3) the use of dots or commas placed before the expression affected, or at the close of such an expression, or (still more commonly) placed both before and after; (4) the use of parentheses (round parentheses or brackets or braces). Of these devices the parentheses were the slowest to find wide adop- tion in all countries, but now they have fairly won their place in competition with the horizontal bar or vinculum. Parentheses pre- vailed for typographical reasons. Other things being equal, there is a preference for symbols which proceed in orderly fashion as do the letters in ordinary printing, without the placing of signs in high or low positions that would break a line into two or more sublines. A vincu- lum at once necessitates two terraces of type, the setting of which calls for more time and greater technical skill. At the present time 1 H. Pitot, Memoires de V academic d. scien., ann6e 1724 (Paris, 1726). Taken from H. W ieleitner, Geschichte der Mathematik, 2. Teil, 2. Halfte (Berlin und Leipzig, 1921), p. 92. To what extent the letter x has been incorporated in mathematical language is illustrated by the French expression Strefort en x, which means "being strong in mathematics." In the same way, fate d x means "a mathematical head." The French give an amusing "demonstration" that old men who were tbte d x never were pressed into military service so as to have been conscripts. For, if they were conscripts, they would now be ex-conscripts. Expressed in symbols we would have Ox = ear-conscript. Dividing both sides by x gives 0=e-conscript. Dividing now by e yields conscript =-. According to this, the conscript would be la tbte assurSe (i.e., over e, or, the head assured against casualty), which is absurd. AGGREGATION 385 the introducing of typesetting machines and the great cost of type- setting by hand operate against a double or multiple line notation. The dots have not generally prevailed in the marking of aggregation for the reason, no doubt, that there was danger of confusion since dots are used in many other symbolisms those for multiplication, division, ratio, decimal fractions, time-derivatives, marking a number into periods of two or three digits, etc. 343. Aggregation expressed by letters. The expression of aggrega- tion by the use of letters serving as abbreviations of words expressing aggregation is not quite as old as the use of horizontal bars, but it is more common in works of the sixteenth century. The need of marking the aggregation of terms arose most frequently in the treatment of radicals. Thus Pacioli, in his Summa of 1494 and 1523, employs v (vniversale) in marking the root of a binomial or polynomial ( 135). This and two additional abbreviations occur in Cardan ( 141). The German manuscript of Andreas Alexander (1524) contains the letters cs for communis (325); Chr. Rudolff sometimes used the word "collect," as in "i/ des collects 17+1/208" to designate > / 17+T/208. 1 J. Scheubel adopted Ha. col. ( 159). S. Stevin, Fr. Vieta, and A. Romanus wrote bin., or bino., or binomia, trinom., or similar abbrevia- tions ( 320). The u or v is found again in Pedro Nunez (who uses also L for "ligature"), 2 Leonard and Thomas Digges ( 334), in J. R. Brasser 3 who in 1663 lets v signify "universal radix" and writes "#l/.8 -T- i/45" to represent ^8 1/45. W. Oughtred sometimes wrote 1/u or i/b ( 183, 334). In 1685 John Wallis 4 explains the notations 1/6:2+1/3, i/r:2-i/3, i/u:2i/3, 1/2 1/3, i/:2l/3, where b means "binomial," u "universal," r "residual," and sometimes uses redundant forms like \/b : i/5+ 1 : . 344. Aggregation expressed by horizontal bars or vinculums. The use ef the horizontal bar to express the aggregation of terms goes back to the time of Nicolas Chuquet who in his manuscript (1484) under- lines the parts affected ( 130). We have seen that the same idea is followed by the German Andreas Alexander ( 325) in a manuscript of 1545, and by the Italian Raffaele Bombelli in the manuscript edition 1 J. Tropfke, op. tit., Vol. II (1921), p. 150. 2 Pedro Nufiez, Libra de algebra en arithmetica y geometria (Anvers, 1507), fol. 52. 1 J. R. Brasser, Regula of Algebra (Amsterdam, 1663), p. 27. * John Wallis, Treatise of Algebra (London, 1685), p. 109, 110. The uso of letters for aggregation practically disappeared in the seventeenth century. 386 A HISTORY OF MATHEMATICAL NOTATIONS of his algebra (about 1550) where he wrote ^2 + V 121 in this manner i 1 R*[2 . p . R[Q m 121]] ; parentheses were used and, in addition, vinculums were drawn underneath to indicate the range of the paren- theses. The employment of a long horizontal brace in connection with the radical sign was introduced by Thomas Harriot 2 in 1631; he expresses aggregation thus: \/ccc+i/cccccc-~bbbbbb. This notation may, perhaps, have suggested to Descartes his new radical sym- bolism of 1637. Before that date, Descartes had used dots in the man- ner of Stif el and Van Ceulen. He wrote 8 1/ . 2 - >/2 . f or 1/2 - 1/2. He attaches the vinculum to the radical sign j/ and writes V / a 2 +b z ) V -|a+l/Jaa+66, and in case of cube roots *C. %q+Vlgg 27p 3 . Descartes does not use parentheses in his Geometric. Descartes uses the horizontal bar only in connection with the radical sign. Its general use for aggregation is due to Fr, van Schooten, who, in edit- ing Vieta's collected works in 1646, discarded parentheses and placed a horizontal bar above the parts affected. Thus Van Schooten's "B in D quad.+B in D" means B(D*+BD). Vieta 4 himself in 1593 had written this expression differently, namely, in this manner: " . (D. quadratwn " B in < , o . ~ \+B in D B. Cavalieri in his Geometria indivisibilibae and in his Exercitationes geometriae sex (1647) uses the vinculum in this manner, AB, to indi- cate that the two letters A and B are not to be taken separately, but conjointly, so as to represent a straight line, drawn from the point A to the point B. Descartes' and Van Schooten's stressing the use of the vinculum led to its adoption by J. Prestet in his popular text, Siemens des Mathtmatiques (Paris, 1675). In an account of Rolle 5 the cube root is to be taken of 2+#.-121, i.e., of 2+1/-121. G. W. Leibniz 6 in a 1 See E. Bertolotti in Scientia, Vol. XXXIII (1923), p. 391 n. 2 Thomas Harriot, Arlis analyticae praxis (London, 1631), p. 100. 3 R. Descartes, (Euvres (6d. Ch. Adam et P. Tannery), Vol. X (Paris, 1908), p. 286 f., also p. 247, 248. 4 See J. Tropfke, op. tit., Vol. II (1921), p. 30. 5 Journal des Sgavans de Tan 1683 (Amsterdam, 1709), p. 97. 8 G. W. Leibniz* letter to D. Oldenburgh, Feb. 3, 1672-73, printed in J. Collin's Commercium epistolicum (1712). AGGREGATION 387 letter of 1672 uses expressions like aco6co6oocc/>&<x>c</>cood, where co signifies "difference." Occasionally he uses the vinculum until about 1708, though usually he prefers round parentheses. In 1708 Leibniz' preference for round parentheses ( 197) is indicated by a writer in the Ada eruditorum. Joh. (1) Bernoulli, in his Lectiones de calculo differ- entialium, uses vinculums but no parentheses. 1 345. In England the notations of W. Oughtred, Thomas Harriot, John Wallis, and Isaac Barrow tended to retard the immediate intro- duction of the vinculum. But it was used freely by John Kersey (1673) 2 who wrote y (2) : |r \/\rr s : and by Newton, as, for in- stance, in his letter to D. Oldenburgh of June 13, 1676, where he gives the binomial formula as the expansion of P+PQ\ n . In his De Analysi per Aequationes numero terminorum Infinitas, Newton writes 3 12x?/-fl7 = to represent {[(y4)y 5]y12}y+17 = 0. This notation was adopted by Edmund Halley, 4 David Gregory, and John Craig; it had a firm foothold in England at the close of the seventeenth century. During the eighteenth century it was the regular symbol of aggregation in England and France; it took the place very largely of the parentheses which are in vogue in our day. The vincu- lum appears to the exclusion of parentheses in the Geometria organica (1720) of Colin Maclaurin, in the Elements of Algebra of Nicholas Saunderson (Vol. I, 1741), in the Treatise of Algebra (2d ed.; London, 1756) of Maclaurin. Likewise, in Thomas Simpson's Mathematical Dissertations (1743) and in the 1769 London edition of Isaac Newton's Universal Arithmetick (translated by Ralphson and revised by Cunn), vinculums are used and parentheses do not occur. Some use of the vinculum was made nearly everywhere during the eighteenth century, especially in connection with the radical sign -j/, so as to produce \/ . This last form has maintained its place down to the present time. However, there are eighteenth-century writers who avoid the vincu- lum altogether even in connection with the radical sign, and use 1 The Johannis (1) Bernoulli! Lectiones de calculo differentialium, which re- mained in manuscript until 1922, when it was published by Paul Schafheitlin in Verhandlungen der Naturforschenden Gesellschaft in Basel, Vol. XXXIV (1922). 2 John Kersey, Algebra (London, 1673), p. 55. 3 Commercium epistolicum (6d. Biot et Lcfort; Paris, 1856), p. 63. 4 Philosophical Transactions (London), Vol. XV-XVI (1684-91), p. 393; Vol. XIX (1695-97), p. 60, 645, 709. 388 A HISTORY OF MATHEMATICAL NOTATIONS parentheses exclusively. Among these are Poleni (1729), 1 Cramer (1750) , 2 and Cossali (1797) . 3 346. There was considerable vacillation on the use of the vinculum in designating the square root of minus unity. Some authors wrote \/ 1; others wrote I/ 1 or j/( !) For example, >/ 1 was the designation adopted by J. Wallis, 4 J. d'Alembert, 6 1. A. Segner, 6 C. A. Vandermonde, 7 A. Fontaine. 8 Odd in appearance is an expression of Euler, 9 v / (2 v / -l-4). But i/(-l) was preferred by Du S<5jour 10 in 1768 and by Waring 11 in 1782; j/-l by Laplace 12 in 1810. 347. It is not surprising that, in times when a notation was passing out and another one taking its place, cases should arise where both are used, causing redundancy. For example, J. Stampioen in Holland sometimes expresses aggregation of a set of terms by three notations, any one of which would have been sufficient; he writes 13 in 1640, I/. (aaa-\-6aab+9bba), where the dot, the parentheses, and the vincu- lum appear; John Craig 14 writes V / 2ay y 2 : and v : V / 6a 4 fa 2 , where the colon is the old Oughtredian sign of aggregation, which is here superfluous, because of the vinculum. Tautology in notation is found in Edward Cocker 15 in expressions like ^aa+bb, V :c+\bb \b, and 1 loannis Poleni, Epistolarvm mathematicarvm fascicvlvs (Padua, 1729). 2 Gabriel Cramer, L' Analyse des lignes courbes algebriques (Geneva, 1750). 3 Pietro Cossali, Origini ... dell'algebra, Vol. I (Parma, 1797). 4 John Wallis, Treatise of Algebra (London, 1685), p. 266. 6 J. d'Alembert in Histoire de Facadtmie r. des sciences, anne*e 1745 (Paris, 1749), p. 383. 6 1. A. Segner, Cursus mathemalici, Pars IV (Halle, 1763), p. 44. 7 C. A. Vandermonde in op. cit., anne*e 1771 (Paris, 1774), p. 385. 8 A. Fontaine, ibid., annee 1747 (Paris, 1752), p. 607. 9 L. Euler in Histoire de Vacademie r. d. sciences et des belles lettres, ann6e 1749 (Berlin, 1751), p. 228. 10 Du S6jour, ibid. (1768; Paris, 1770), p. 207. 11 E. Waring, Meditationes algebraicae (Cambridge; 3d ed., 1782), p. xxxvl, etc. 13 P. S. Laplace in Memoires d. Vacademie r. d. sciences, anne*e 1817 (Paris, 1819), p. 153. 11 /. /. Stampionii Wis-Konsligh ende Reden-Maetigh Bewijs (The Hague, 1640), p. 7. 14 John Craig, Philosophical Transactions, Vol. XIX (London, 1695-97), p. 709. 15 Cockers Artificial Arithmelick Composed by Edward Cocker Pe- rused, corrected and published by John Hawkins (London, 1702) ['To the Reader," 1684], p. 368, 375. AGGREGATION 389 a few times in John Wallis. 1 In the Ada eruditorum (1709), page 327, one finds n7/V / a = ||/[(x nna) 3 ], where the [ ] makes, we believe, its first appearance in this journal, but does so as a redundant symbol. 348. Aggregation expressed by dots. The denoting of aggregation by placing a dot before the expression affected is first encountered in Christoff Rudolff ( 148). It is found next in the Arithmetica Integra of M. Stifel, who sometimes places a dot also at the end. He writes 2 1/2.12+i/z 6+.1/2- 12-1/2 6 for our V12+V / 6+^12-l6; also 1/0.144-6+1/^.144-6 for 1/114^+1/144^6. In 1605 C. Di- buadius 3 writes v'.2-i/.2+j/.2+v'.2+j/.2+v / 2 as the side of a regular polygon of 128 sides inscribed in a circle of unit radius, i.e., \2-\2+V2+'V2+ v/ 2+T/2 (see also 332). It must be ad- mitted that this old notation is simpler than the modern. In SnelPs translation 4 into Latin (1610) of Ludolph van Ceulen's work on the circle is given the same notation, i/.2+i/.2 1/.2 i/.2+i/2| 1/2J. In SnelFs 1615 translation 6 into Latin of Ludolph's arithmetic and geometry is given the number i/.2 1/.2J + 1/1J which, when divided by j/.2+j/.2+f/lJ, gives the quotient i/5+l-j/.5+ 1/20. The Swiss Joh. Arduser 6 in 1646 writes i/.2-f-i/-2+i/.2+ I/. 2+ 1/2+ 1/. 2+ 1/. 2+ 1/3, etc., as the side of an inscribed poly- gon of 768 sides, where -f- means "minus." The substitution of two dots (the colon) in the place of the single dot was effected by Oughtred in the 1631 and later editions of his Clavis mathematicae. With him this change became necessary when he adopted the single dot as the sign of ratio. He wrote ordinarily l/q:BC q BA q : for ^BC 2 BA 2 , placing colons before and after the terms to be aggregated ( 181). 7 1 John Wallis, Treatise of Algebra (London, 1685), p. 133. 2 M. Stifel, Arithmetica integra (Nurnbcrg, 1544), fol. 135t>. See J. Tropfke, op. cit., Vol. Ill (Leipzig, 1922), p. 131. 3 C. Dibvadii in arithmeticam irrationalivm Evclidis decimo elernentorum libro (Arnhem, 1605). 4 Willebrordus Snellius, Lvdolphi a Cevlen de Circvlo et adscriptis liber ...6 vernaculo Latina fecit ... (Leyden, 1610), p. 1, 5. 6 Fvndamenta arithmetica et geometrica. ... Lvdolpho a Cevlen , ... in Lalinum translata a Wil. Sn. (Leyden, 1615), p. 27, 6 Joh. Arduser, Geomelriae theoricae et practicae XII libri (Zurich, 1646), fol. 1816. 7 W Oughtred, Clavis mathematicae (1652), p. 104. 390 A HISTORY OF MATHEMATICAL NOTATIONS Sometimes, when all the terms to the end of an expression are to be aggregated, the closing colon is omitted. In rare instances the opening colon is missing. A few times in the 1694 English edition, dots take the place of the colon. Oughtred's colons were widely used in England. As late as 1670 and 1693 John Wallis 1 writes i/:5-2v / 3:. It occurs in Edward Cocker's 2 arithmetic of 1684, Jonas Moore's arithmetic 3 of 1688, where C:A+E means the cube of (A-\-E). James Bernoulli 4 gives in 1689 |/:a+i/:a+i/:a+i/:a-fj/:a-K etc. These methods of denoting aggregation practically disappeared at the beginning of the eighteenth century, but in more recent time they have been reintroduced. Thus, R. Carmichacl 5 writes in his Calculus of Operations: "D. uv~u. Dv-\-Du. v." G, Peano has made the proposal to employ points as well as parentheses. 6 He lets a. be be identical with a(6c), aibc.d with a[(bc)d] y ab.cdie.fg .'. hk.l with {[(ab)(cd)] 349. Aggregation expressed by commas. An attempt on the part of Hdrigone ( 189) and Leibniz to give the comma the force of a symbol of aggregation, somewhat similar to Rudolffs, Stifel's, and van Ceulen's previous use of the dot and Oughtred's use of the colon, was not successful. In 1702 Leibniz 7 writes c b } I for (c b)l, and c 6, d 6, I for (c b)(db)L In 1709 a reviewer 8 in the Ada eruditorum represents (m-[ml])x (m ~' l) * m by (m,:m l)x w ~ 1 ' :m , a designation somewhat simpler than our modern form. 350. Aggregation expressed by parenthesis is found in rare in- stances as early as the sixteenth century. Parentheses present com- paratively no special difficulties to the typesetter. Nevertheless, it took over two centuries before they met with general adoption as mathematical symbols. Perhaps the fact that they were used quite extensively as purely rhetorical symbols in ordinary writing helped to 1 John Wallis in Philosophical Transactions, Vol. V (London, for the year 1670), p. 2203; Treatise of Algebra (London, 1685), p. 109; Latin ed. (1693), p. 120. 2 Cocker's Artificial Arithmetick .... perused .... by John Hawkins (Lon- don, 1684), p. 405. 3 Moore's Arithmetick: in Four Books (London, 1688; 3d ed.), Book IV, p. 425. 4 Positiones arithmeticae de seriebvs infinilis .... Jacobo Bernoulli (Basel, 1689). 6 R. Carmichael, Der Operationscalcul, deutsch von C. H. Schnuse (Braun- schweig, 1857), p. 16. 6 G. Peano, Formulaire mathemalique, fidition de Fan 1902-3 (Turin, 1903), p. 4. 7 G. W. Leibniz in Ada eruditorum (1702), p. 212. 8 Reviewer in ibid. (1709), p. 230. See also p. 180. AGGREGATION 391 retard their general adoption as mathematical symbols. John Wallis, for example, used parentheses very extensively as symbols containing parenthetical rhetorical statements, but made practically no use of them as symbols in algebra. As a rhetorical sign to inclose an auxiliary or parenthetical state- ment parentheses are found in Newton's De analysi per equaliones numero lerminorum infinilas, as given by John Collins in the Com- merdum epislolicum (1712). In 1740 De Gua 1 wrote equations in the running text and inclosed them in parentheses; he wrote, for example, "... seroit (7 a 3x dx = 3l/2az xx dx) et oil Tare de cercle. ..." English mathematicians adhered to the use of vinculums, and of colons placed before and after a polynomial, more tenaciously than did the French; while even the French were more disposed to stress their use than were Leibniz and Euler. It was Leibniz, the younger Bernoullis, and Euler who formed the habit of employing parentheses more freely and to resort to the vinculum less freely than did other mathematicians of their day. The straight line, as a sign of aggrega- tion, is older than the parenthesis. We have seen that Chuquet, in his Triparly of 1484, underlined the terms that were to be taken together. 351. Early occurrence of parentheses. Brackets 2 are found in the manuscript edition of R. Bombelli's Algebra (about 1550) in the expressions like &[2mR[Om.l2l]] which stands for ^2-l/-121. In the printed edition of 1572 an inverted capital letter L was employed to express radix legata; see the facsimile reproduction (Fig. 50). Michael Stifel does not use parentheses as signs of aggregation in his printed works, but in one of his handwritten marginal notes 3 occurs the following: ". . . . faciant aggregatum (12 j/44) quod sumptum cum (v/44 2) facial 10" (i.e., "....One obtains the aggregate (12 -J/44), which added to (j/44 2) makes 10"). It is our opinion that these parentheses are punctuation marks, rather than mathe- matical symbols; signs of aggregation are not needed here. In the 1593 edition of F. Vieta's Zetelica, published in Turin, occur braces and brackets ( 177) sometimes as open parentheses, at other times as closed ones. In Vieta's collected works, edited by Fr. van Schooten 1 Jean Paul de Gua de Malves, Usages de V analyse de Descartes (Paris, 1740), p. 302. 2 See E. Bortolotti in Scientia, Vol. XXXIII (1923), p. 390. 3 E. Hoppe, "Michael Stifels handschriftlicher Nachlass," Mitteilungen Math. Gesellschaft Hamburg, III (1900), p. 420. See J. Tropfke, op. cii. 9 Vol. II (2d ed., 1921), p. 28, n. 114. 392 A HISTORY OF MATHEMATICAL NOTATIONS in 1646, practically all parentheses are displaced by vinculums. How- ever, in J. L. de Vaulezard's translation 1 into French of Vieta's Zetetica round parentheses are employed. Round parentheses are en- countered in Tartaglia, 2 Cardan (but only once in his Ars Magnci)? Clavius (see Fig. 66), Errard de Bar-le-Duc, 4 Follinus, 5 Girard, 6 Norwood, 7 Hume, 8 Stampioen, Henrion, Jacobo de Billy, 9 Renaldini 10 and Foster. 11 This is a fairly representative group of writers using parentheses, in a limited degree; there are in this group Italians, Ger- mans, Dutch, French, English. And yet the mathematicians of none of the countries represented in this group adopted the general use of parentheses at that time. One reason for this failure lies in the fact that the vinculum, and some of the other devices for expressing ag- gregation, served their purpose very well. In those days when machine processes in printing were not in vogue, and when typesetting was done by hand, it was less essential than it is now that symbols should, in orderly fashion, follow each other in a line. If one or more vincu- lums were to be placed above a given polynomial, such a demand upon the printer was less serious in those days than it is at the present time. 1 J. L. de Vaulezard's Zetettques de F. Viete (Paris, 1630), p. 218. Reference taken from the Encyclopedic d. scien. math., Tom I, Vol. I, p. 28. 2 N. Tartaglia, General trattato di numeri e misure (Venice), Vol. II (1556), fol. 1676, 1696, 1706, 1746, 177a, etc., in expressions like "&v.(ft 28 men R 10)" for VV28-I/10; f l- 168& " men (22 men #6" for -(22 -1/6), only the opening part being used. See G. Enestrom in Bibliotheca mathematica (3d ser.), Vol. VII (1906-7), p. 296. Similarly, in La Quarta Parte del general trattato (1560), fol. 40#, he regularly omits the second part of the parenthesis when occurring on the margin, but in the running text both parts occur usually. 3 H. Cardano, Ars magna, as printed in Opera, Vol. IV (1663), fol. 438. 4 1. Errard de Bar-le-Duc, La geometric et practique generate d'icelle (3d ed.; revue* par D. H. P. E. M.; Paris, 1619), p. 216. 6 Hermann Follinus, Algebra sive liber de rebus occvltis (Cologne ; 1622), p. 157. 8 A. Girard, Invention nouvclle en Valgebre (Amsterdam, 1629), p. 17. 7 R. Norwood, Trigonometric (London, 1631), Book I, p. 30. 8 Jac. Humius, Traite de Valgebre (Paris, 1635). 9 Jacobo de Billy, Novae geometriae clavis algebra (Paris, 1643), p. 157; also in an Abridgement of the Precepts of Algebra (written in French by James de Billy; London, 1659), p. 346. 10 Carlo Renaldini, Opus algebricum (1644; enlarged edition, 1665). Taken from Ch. Hutton, Tracts on Mathematical and Philosophical Subjects, Vol. II (1812), p. 297. 11 Samuel Foster, Miscellanies: or Mathematical Lucubrations (London, 1659), p. 7. AGGREGATION 393 And so it happened that in the second half of the seventeenth century, parentheses occur in algebra less frequently than during the first half of that century. However, voices in their favor are heard. The Dutch writer, J. J. Blassiere, 1 explained in 1770 the three nota- tions (2a+56)(3a-4&), (2a+56)X(3a-46), and 2a+MX3a^4b, and remarked: "Mais comme la premiere manire de les enfermer entre des Parentheses, est la moins sujette a erreur, nous nous en servirons dans la suite." E. Waring in 1762 2 uses the vinculum but no parentheses; in 1782 3 he employs parentheses and vinculurns inter- changeably. Before the eighteenth century parentheses hardly ever occur in the Philosophical Transactions of London, in the publications of the Paris Academy of Sciences, in the A eta eruditorum published in Leipzig. But with the beginning of the eighteenth century, paren- theses do appear. In the Ada eruditorum, Carre 4 of Paris uses them in 1701, G. W. Leibniz 5 in 1702, a reviewer of Gabriele Manfredi 6 in 1708. Then comes in 1708 ( 197) the statement of policy 7 in the Acta eruditorum in favor of the Leibnizian symbols, so that "in place of l/aa+bb we write i/(aa+bb) and for aa+bbXc we write aa+bb,c .... we shall designate aa+bb m by (aa-\-bb) m : whence Vaa+bb will m/ be =(aa+bb) 1:m and v aa+bb n =(aa+bb) n:m . Indeed, we do not doubt that all mathematicians reading these Acta recognize the pre- eminence of Mr. Leibniz' symbolism and agree with us in regard to it." From now on round parentheses appear frequently in the Acta eruditorum. In 1709 square brackets make their appearance. 8 In the Philosophical Transactions of London 9 one of the first appearances of parentheses was in an article by the Frenchman P. L. Maupertuis in 1731, while in the Histoire de Vacademie royale des sciences in Paris, 10 1 J. J. Blassiere, Institution du calcul numerique et litteral. (a la Haye, 1770), 2. Partie, p. 27. 2 E. Waring, Miscellanea analytica (Cambridge, 1762). 3 E. Waring, Meditationes algebraicae (Cambridge; 3d ed., 1782). <L. Carre* in Acta eruditorum (1701), p. 281. G. W. Leibniz, ibid. (1702), p. 219. 6 Gabriel Manfredi, ibid. (1708), p. 268. 7 Ibid. (1708), p. 271. s Ibid. (1709), p. 327. 9 P. L. Maupertuis in Philosophical Transactions, for 1731-32, Vol. XXXVII (London), p. 245. 10 Johann II Bernoulli, Histoire de Vacadtmie royale des sciences, ann6e 1732 (Paris, 1735), p. 240 ff. 394 A HISTORY OF MATHEMATICAL NOTATIONS Johann (John) Bernoulli of Bale first used parentheses and brackets in the volume for the year 1732. In the volumes of the Petrograd Academy, J. Hermann 1 uses parentheses, in the first volume, for the year 1726; in the third volume, for the year 1728, L. Euler 2 and Daniel Bernoulli used round parentheses and brackets. 352. The constant use of parentheses in the stream of articles from the pen of Euler that appeared during the eighteenth century con- tributed vastly toward accustoming mathematicians to their use. Some of his articles present an odd appearance from the fact that the closing part of a round parenthesis is much larger than the opening ?t Z \ part, 1 as in (1 ) (1 ). Daniel Bernoulli 4 in 1753 uses round 7T 7T S j parentheses and brackets in the same expression while T. U. T, Aepinus 5 and later Euler use two types of round parentheses of this sort, C(P+y)(M-l)+AM3. In the publications of the Paris Academy, parentheses are used by Johann Bernoulli (both round and square ones), 6 A. C. Clairaut, 7 P. L. Maupertuis, 8 F. Nicole, 9 Ch. de Montigny, 10 Le Marquis de Courtivron, 11 J. d'Alembert, 12 N. C. de Condorcet, 13 J. Lagrange. 14 These illustrations show that about the middle of the eighteenth century parentheses were making vigorous inroads upon the territory previously occupied in France by vincu- lums almost exclusively. 1 J. Hermann, Commentarii academiae sdentiarum imperialis Petropolitanae, Tomus I ad annum 1726 (Petropoli, 1728), p. 15. 2 Ibid., Tomus III (1728; Petropoli, 1732), p. 114, 221. 3 L. Euler in Miscellanea Berolinensia, Vol. VII (Berlin, 1743), p. 93, 95, 97, 139, 177. 4 D. Bernoulli in Histoire de I'academie r. des sciences et belles lettres, anne*e 1753 (Berlin, 1755), p. 175. 5 Aepinus in ibid., anne"e 1751 (Berlin, 1753), p. 375; anne*e 1757 (Berlin, 1759), p. 308-21. 6 Histoire de Vacademie r. des sciences, anne*e 1732 (Paris, 1735), p. 240, 257. 7 Ibid., ann&j 1732, p. 385, 387. 8 Ibid., ann6e 1732, p. 444. 9 Ibid., anne*e 1737 (Paris, 1740), "Me*moires," p. 64; also ann& 1741 (Paris, 1744), p. 36. 10 Ibid., ann6e 1741, p. 282. 11 Ibid., ann6e 1744 (Paris, 1748), p. 406. 12 Ibid., ann6e 1745 (Paris, 1749), p. 369, 380. 13 Ibid., ann6e 1769 (Paris, 1772), p. 211. " Ibid., anne*e 1774 (Paris, 1778), p. 103. AGGREGATION 395 353. Terms in an aggregate placed in a vertical column. The em- ployment of a brace to indicate the sum of coefficients or factors placed in a column was in vogue with Vieta (176), Descartes, and many other writers. Descartes in 1637 used a single brace, 1 as in cc or a vertical bar 2 as in cc zz ace JLa* ,+*>* o. \aacc Wallis 3 in 1685 puts the equation aaa-{-baa-\-cca = ddd, where a is the unknown, also in the form 1 aaa + b aa + CC 1 Sometimes terms containing the same power of x were written in a column without indicating the common factor or the use of symbols of aggregation; thus, John Wallis 4 writes in 1685, aaa + baa + bca = + bed +caa bda daacda Giovanni Poleni 5 writes in 1729, y* + xxy 4 2ax 3 yy + aax* = 2axy 4 + aaxxyy - aay 4 The use of braces for the combination of terms arranged in col- umns has passed away, except perhaps in recording the most unusual algebraic expressions. The tendency has been, whenever possible, to discourage symbolism spreading out vertically as well as horizontally. Modern printing encourages progression line by line. 354. Marking binomial coefficients. In the writing of the factors in binomial coefficients and in factorial expressions much diversity of practice prevailed during the eighteenth century, on the matter of 1 Descartes, (Euvres (e*d. Acfain et Tannery), Vol. VI, p. 450. 2 Ibid. 3 John Wallis, Treatise of Algebra (London, 1685), p. 160. 4 John Wallis, op. ciL, p. 153. 6 Joannis Poleni, Epistolarvm mathematicarvm fascicvlvs (Padua, 1729) (no pagination). 396 A HISTORY OF MATHEMATICAL NOTATIONS the priority of operations indicated by + and , over the operations of multiplication marked by and X. Inn.n 1-w 2 or nXn IX n 2, or n,n 1, n2, it was understood very generally that the sub- tractions are performed first, the multiplications later, a practice con- trary to that ordinarily followed at that time. In other words, these expressions meant n(n l)(n 2). Other writers used parentheses or vinculums, which removed all inconsistency and ambiguity. Nothing was explicitly set forth by early writers which would attach different meanings to nn and n-n or nXn. And yet, nn l*n 2 was not the same as nn In 2. Consecutive dots or crosses tacitly conveyed the idea that what lies between two of them must be aggregated as if it were inclosed in a parenthesis. Some looseness in notation occurs even before general binomial coefficients were introduced. Isaac Barrow 1 wrote "L-MX :R+S" for (L-M)(fl+S), where the colon designated aggregation, but it was not clear that L M, as well as R+S, were to be aggregated. In a manuscript of Leibniz 2 one finds the number of combinations of n things, taken k at a time, given in the form -2, etc., n k+1 This diversity in notation continued from the seventeenth down into the nineteenth century. Thus, Major Edward Thornycroft (1704) 3 writes mXrn iXm 2Xm 3, etc. A writer 4 in the Acta eruditorum gives the expression n,nl. Another writer 5 gives (n,n-"l,n-2)^ Leibniz > 6 notation, as described in 1710 ( 198), con- Z,6 tains e*e l*e 2 for e(el)(e2). Johann Bernoulli 7 writes nn lw 2. This same notation is used by Jakob (James) Bernoulli 8 in a 1 Isaac Barrow, Lectiones mathematical, Lect. XXV, Probl. VII. See also Probl. VIII. 2 D. Mahnke, Bibliotheca mathematica (3d ser.), Vol. XIII (1912-13), p. 35. See also Leibnizens Mathematische Schriften, Vol. VII (1863), p. 101. 3 E. Thornycroft in Philosophical Transactions, Vol. XXIV (London, 1704-5), p. 1963. *Acto eruditorum (Leipzig, 1708), p. 269. *Ibid., Suppl., Tome IV (1711), p. 160. t 6 M iscellanea Berolinensia (Berlin, 1710), p. 161. 7 Johann Bernoulli in Acta eruditorum (1712), p. 276. 8 Jakob Bernoulli, Ars Conjectandi (Basel, 1713), p. 99. AGGREGATION 397 posthumous publication, by F. Nicole 1 who uses x+nx+2n>x+3n > etc., by Stirling 2 in 1730, by Cramer 3 who writes in a letter to J. Stirling a^a+6a+26, by Nicolaus Bernoulli 4 in a letter to Stirling rr+&r+2&. ... by Daniel Bernoulli 5 Z 1-Z 2, by Lambert 6 4m ! 4m 2, and by Konig 7 n*n 5n 6n 7. Euler 8 in 1764 employs in the same article two notations: one, n 5n 6n 7; the other, w(n l)(n-2). Condorcet 9 has n+2Xw+l. Hindcnburg 10 of Got- tingen uses round parentheses and brackets, nevertheless he writes binomial factors thus, m-ml>m 2 . . . .m s+l. Segner 11 and Ferroni 12 write n *n 1 -n 2. Cossali 13 writes 4X 2= 8. As late as 1811 A. M. Legendre 14 has w-n 1-n 2 .... 1. On the other hand, F. Nicole, 15 who in 1717 avoided vinculums, writes in 1723, a;n+w-x+2n, etc. Stirling 16 in 1730 adopts z\ *z 2. De Moivre 17 in 1730 likewise writes mpXm qX m s, etc. Similar- ly, Dodson, 18 n*n-~ ln 2, and the Frenchman F. de Lalande, 19 1 Nicole in Histoire de I'acad6mie r. des sciences, annec 1717 (Paris, 1719), "Memoires," p. 9. 2 J. Stirling, Methodus dijjerentialis (London, 1730), p. 9. 3 Ch. Tweedic, James Stirling (Oxford, 1922), p. 121. 4 Op. tit., p. 144. 5 Daniel I. Bernoulli, "Notationes de aequationibus," Comment. Acad. Pelrop., Tome V (1738), p. 72. 6 J. H. Lambert, Observaliones in Ada Helvetica, Vol. III. 7 S. Konig, Histoire de V academic r. des sciences et des belles lettres, ann6e 1749 (Berlin, 1751), p. 189. 8 L. Euler, op. cit., annee 1764 (Berlin, 1760), p. 195, 225. 9 N. C. de Condorcet in Histoire de I'acadcmie r. des sciences, ann6e 1770 (Paris, 1773), p. 152. 10 Carl Fricdrich Hindenburg, Infinitinomii dignatum leges . . . . ac formulae (Gottingen, 1779), p. 30. 11 J. A. de Segner, Cursus mathemalici, pars II (Halle, 1768), p. 190. 12 P. Ferroni, Magniludinum exponentialium .... theoria (Florence, 1782), p. 29. 13 Pietro Cossali, Origine, trasporto in Italia ... dell 1 algebra, Vol. I (Parma, 1797), p. 260. 14 A. M. Legendre, Exercices de calcul integral, Tome I (Paris, 1811), p. 277. 18 Histoire de Vacademie r. des sciences, ann6e 1723 (Paris, 1753), "Memoires," p. 21. 16 James Stirling, Methodus differ entialis (London, 1730), p. 6. 17 Abraham de Moivre, Miscellanea analytica de seriebus (London, 1730), p. 4. 18 James Dodson, Mathematical Repository, Vol. I (London, 1748), p. 238. 19 F. de Lalande in Histoire de I'academie r. des sciences, anne*e 1761 (Paris, 1763), p. 127. 308 A HISTORY OF MATHEMATICAL NOTATIONS m(m+l).(m+2). In Lagrange 1 we encounter in 1772 the strictly modern form (m+l)(m+2)(m+3), . . . . , in Laplace 2 in 1778 the form (t-l).(i-2) ____ (i-r+1). The omission of parentheses unnecessarily aggravates the inter- pretation of elementary algebraic expressions, such as are given by Kirkman, 3 viz., -3 = 3X-1 for -3 = 3X(-1), wXn for (- w )(-n). 355. Special uses of parentheses. A use of round parentheses and brackets which is not strictly for the designation of aggregation is found in Cramer 4 and some of his followers. Cramer in 1750 writes two equations involving the variables x and y thus: A ____ x'-[-l]x- l +[l*\x*-*-[l*]x-*+ &c B ____ (0)z+(lK+(2)z 2 +(3)z 3 + &c ..... where 1, I 2 , I 3 , . . . . , within the brackets of equation A do not mean powers of unity, but the coefficients of x, which are rational functions of y. The figures 0, 1, 2, 3, in B are likewise coefficients of x and func- tions of y. In the further use of this notation, (02) is made to repre- sent the product of (0) and (2); (30) the product of (3) and (0), etc. Cramer's notation is used in Italy by Cossali 5 in 1799. Special uses of parentheses occur in more recent time. Thus W. F. Sheppard 8 in 1912 writes (n,r) for n(w-l) ____ (n-r+l)/r! [n,r] for n(n+l) ____ (n+r-l)/r! for (n-s)(w 356. A star to mark the absence of terms. We find it convenient to discuss this topic at this time. Ren Descartes, in La Geometrie (1637), arranges the terms of an algebraic equation according to the descending order of the powers of the unknown quantity x, y y or z. If any power of the unknown below the highest in the equation is 1 J. Lagrange in ibid., ann6e 1772, Part I (Paris, 1775), "M6moires," p. 523. 2 P. S. Laplace in ibid., ann6e 1778 (Paris, 1781), p. 237. 3 T. P. Kirkman, First Mnemonical Lessons in Geometry, Algebra and Trigonom- etry (London, 1852), p. 8, 9. < Gabriel Cramer, Analyse des Lignes cowrbes algebriqu&s (Geneva,* 1750), p. 660. 5 Pietro Cossali, op. tit., Vol. II (Parma, 1799), p. 41. 6 W. F. Sheppard in Fifth International Mathematical Congress, Vol. II, p. 355. AGGREGATION 399 lacking, that fact is indicated by a *, placed where the term would have been. Thus, Descartes writes z 6 a 4 6x = in this manner: 1 e * * * He does not explain why there was need of inserting these stars in the places of the missing terms. But such a need appears to have been felt by him and many other mathematicians of the seventeenth and eighteenth centuries. Not only were the stars retained in later edi- tions of La Geometric, but they were used by some but not all of the leading mathematicians, as well as by many compilers of textbooks. Kinckhuysen 2 writes "x? * * * * &0." Prestet 3 in 1675 writes a 3 **+6 3 , and retains the * in 1689. The star is used by Baker, 4 Varig- non, 5 John Bernoulli, 6 Alexander, 7 A. de Graaf, 8 E. Hallcy. 9 Fr. van Schooten used it not only in his various Latin editions of Descartes 7 Geometry, but also in 1646 in his Conic Sections, 10 where he writes z**-pz+qforz*= -pz+q. InW.Whiston's 11 1707 edition of I. New- ton's Universal Arithmetick one reads aa* bb and the remark ". . . . locis vacuis substituitur nota * ." Raphson's English 1728 edition of the same work also uses the *. Jones 12 uses * in 1706, Reyneau 13 in 1708; Simpson 14 employs it in 1737 and Waring 15 in 1762. De Lagny 16 1 Reno Descartes, La geometric (Leyden, 1637) ; (Euvres de Descartes (e*d. Adam et Tannery), Vol. VI (1903), p. 483. 2 Gerard Kinckhuysen, Algebra ofte Stel-Konst (Haarlem, 1061), p. 59. 3 Elemens des mathematiques (Paris, 1675), Epttre, by J. P.[rostct], p. 23. Nouvcaux elemens des Mathematiques, par Jean Prestet (Paris, 1689), Vol. II, p. 450. 4 Thomas Baker, Geometrical Key (London, 1684), p. 13. 5 Journal des Sgavans, ann6e 1687, Vol. XV (Amsterdam, 1688), p. 459. The star appears in many other places of this Journal. 6 John Bernoulli in Ada eruditorum (1688), p. 324. The symbol appears often in this journal. 7 John Alexander, Synopsis Algebraica ... (Londini, 1693), p. 203. 8 Abraham de Graaf, De Geheele Mathesis (Amsterdam, 1694), p. 259. 9 E. Halley in Philosophical Transactions, Vol. XIX (London, 1695-97), p. 61. 10 Francisci a Schooten, De organica conicarum sectionum (Leyden, 1646), p. 91. 11 Arithmetica universalis (Cambridge, 1707), p. 29. 12 W. Jones, Synopsis palmariorum matheseos (London, 1706), p. 178. "Charles Reyneau, Analyse demontree, Vol. I (Paris, 1708), p. 13, 89. 14 Thomas Simpson, New Treatise of Fluxions (London, 1737), p. 208. > w Edward Waring, Miscellanea Analytica (1762), p. 37. 16 Memoires de Vacademie r. d. sciences. Depuis 1666 jusqu'a 1699, Vol. XI (Paris, 1733), p. 241, 243, 250. 400 A HISTORY OF MATHEMATICAL NOTATIONS employs it in 1733, De Gua 1 in 1741, MacLaurin 2 in his Algebra, and Fenn 3 in his Arithmetic. But with the close of the eighteenth century the feeling that this notation was necessary for the quick understand- ing of elementary algebraic polynomials passed away. In more ad- vanced fields the star is sometimes encountered in more recent authors. Thus, in the treatment of elliptic functions, Weicrstrass 4 used it to mark the absence of a term in an infinite series, as do also Greenhill 5 and Fricke. 6 1 Histoire de I' academic r. d. sciences, anne*e 1741 ( Paris, 1744), p. 476. 2 Colin Maclaurin, Treatise of Algebra (2d 6d.; London, 1756), p. 277. 3 Joseph Fcnn, Universal Arithmetic (Dublin, 1772), p. 33. 4 II. A. Schwarz, Formeln und Lehrsdtze .... nach Vorlesungen des Weier- sirass (Gottingen, 1885), p. 10, 11. 5 A. G. Greenhill, Elliptic Functions (1892), p. 202, 204. 6 R. Fricke, Encyklopadie d. Math. Wissenschaften, Vol. II 2 (Leipzig, 1913), p. 269. IV SYMBOLS IN GEOMETRY (ELEMENTARY PART) A. ORDINARY ELEMENTARY GEOMETRY 357. The symbols sometimes used in geometry may be grouped roughly under three heads: (1) pictographs or pictures representing geometrical concepts, as A representing a triangle; (2) ideographs de- signed especially for geometry, as ^> for "similar"; (3) symbols of elementary algebra, like + and . Early use of pictographs. The use of geometrical drawings goes back at least to the time of Ahmes, but the employment of pictographs in the place of words is first found in Heron's Dioptra. Heron (150 A.D.) wrote /s for triangle, -^ for parallel and parallelogram, also ~gl for parallelogram, r~V for rectangle, Q for circle. 1 Similarly, Pappus (fourth century A.D.) writes O and O for circle, v an d A for triangle, L for right angle, Ji. or == for parallel, D for square. 2 But these were very exceptional uses not regularly adopted by the authors and occur in few manuscripts only. They were not generally known and are not encountered in other mathematical writers for about one thousand years. Paul Tannery calls attention to the use of the symbol D in a medieval manuscript to represent, not a square foot, but a cubic foot; Tannery remarks that this is in accordance with the ancient practice of the Romans. 8 This use of the square is found in the Triparty of Chuquet ( 132) and in the arithmetic of De la Roche. 358. Geometric figures were used in astrology to indicate roughly the relative positions of two heavenly bodies with respect to an ob- server. Thus 6, cP, D, A, H< designated, 4 respectively, conjunction, 1 Notices et extraits des manuscrits de la Bibliothkque imp&riale, Vol. XIX, Part II (Paris, 1858), p. 173. 2 Pappi Alexandrini Cottectionis quae supersuni (ed. F. Hultsch), Vol. Ill, Tome I (Berlin, 1878), p. 126-31. 8 Paul Tannery, M&moires scientifiques, Vol. V (Toulouse and Paris, 1922), p. 73. 4 Kepler says: "Quot sunt igitur aspectus? Vetus astrologia agnoscit t ant urn quinque: conjunctionem (<$), cum radii planetarum binorum in Terram de- scendentes in unam conjunguntur lineam; quod eat veluti principium aspectuum omnium. 2) Oppositionem (cP) cum bini radii sunt ejusdem rectae partes, seu 401 402 A HISTORY OF MATHEMATICAL NOTATIONS opposition, at right angles, at 120, at 60. These signs are repro- duced in Christian Wolff's Mathematisches Lexicon (Leipzig, 1716), page 188. The #, consisting of three bars crossing each other at 60, was used by the Babylonians to indicate degrees. Many of their war carriages are pictured as possessing wheels with six spokes. 1 359. In Plato of Tivoli's translation (middle of twelfth century) of the Liber embadorum by Savasorda who was a Hebrew scholar, at Barcelona, about 1100 A.D., one finds repeatedly the designations abc, ab for arcs of circles. 2 In 1555 the Italian Fr. Maurolycus 8 employs A, D, also %. for hexagon and / .* for pentagon, while in 1575 he also used CU. About half a century later, in 1623, Metius in the Nether- lands exhibits a fondness for pictographs and adopts not only t^, D, but a circle with a horizontal diameter and small drawings represent- ing a sphere, a cube, a tetrahedron, and an octohedron. The last four were never considered seriously for general adoption, for the obvious reason that they were too difficult to draw. In 1634, in France, H6ri- gone's Cursus mathematicus ( 189) exhibited an eruption of symbols, both pictographs and arbitrary signs. Here is the sign < for angle, the usual signs for triangle, square, rectangle, circle, also J for right angle, the Heronic = for parallel, <^> for parallelogram, ^ for arc of circle, f* for segment, for straight line, _L for perpendicular, 5< for pentagon, 6< for hexagon. In England, William Oughtred introduced a vast array of char- acters into mathematics ( 181-85); over forty of them were used in symbolizing the tenth book of Euclid's Elements ( 183, 184), first printed in the 1648 edition of his Clavis mathematicae. Of these sym- bols only three were pictographs, namely, Q for rectangle, D for square, A for triangle ( 184). In the first edition of the Clavis (1631), the Q alone occurs. In the Trigonometria (1657), he employed /. for angle and for angles ( 182), || for parallel occurs in Oughtred's cum duae quartae partes circuli a binis radiis interceptae sunt, id est unus semi- circulus. 3) Tetragonum seu quadratum (D), cum una quarta. 4) Trigonumseu trinum ( A), cum una tertia seu duae sextae. 5) Hexagonum seu sextilem (>(<), cum una sexta." See Kepler, Opera omnia (ed. Ch. Frisch), Vol. VI (1866), p. 490, quoted from "Epitomes astronomiae" (1618). 1 C. Bezold, Ninive und Babylon (1903), p. 23, 54, 62, 124. See also J. Tropfke, op. tit., Vol. I (2d ed., 1921), p. 38. 2 See M. Curtze in Bibliotheca mathematica (3d ser.), Vol. I (1900), p. 327, 328. 3 Frantisci Maurolyd Abbatis Messanensis Opuscula Mathematica (Venice, 1575), p. 107, 134. See also Francisco Maurolyco in Boncompagni's Bulletino, Vol. IX, p. 67. GEOMETRY 403 Opuscula mathematica hactenus inedita (1677), a posthumous work ( 184). Kltigel 1 mentions a cube O as a symbol attached to cubic meas- ure, corresponding to the use of n in square measure. Euclid in his Elements uses lines as symbols for magnitudes, in- cluding numbers, 2 a symbolism which imposed great limitations upon arithmetic, for he does not add lines to squares, nor does he divide a line by another line. 360. Signs for angles. We have already seen that H6rigone adopted < as the sign for angle in 1634. Unfortunately, in 1631, Harriot's Artis analyticae praxis utilized this very symbol for "less than." Harriot's > and < for "greater than" and "less than" were so well chosen, while the sign for "angle" could be easily modified so as to remove the ambiguity, that the change of the symbol for angle was eventually adopted. But < for angle persisted in its appearance, especially during the seventeenth and eighteenth centuries. We find it in W. Leybourn, 3 J. Kersey, 4 E. Hatton, 6 E. Stone, 8 J. Hodgson, 7 D'Alembert's Encyclopedie* Hall and Steven's Euclid* and Th. Reye. 10 John Caswell 11 used the sign g to express "equiangular." A popular modified sign for angle was Z , in which the lower stroke is horizontal and usually somewhat heavier. We have encountered this in Oughtred's Trigonometria (1657), Caswell, 12 Dulaurens, 13 !G. S. Kliigel, Math. Worterbuch, 1. Theil (Leipzig, 1803), art. "Bruch- zeichen." 2 Sec, for instance, Euclid's Elements, Book V; see J. Gow, History of Greek Mathematics (1884), p. 106. 3 William Leybourn, Panorganon: or a Universal Instrument (London, 1672), p. 75. 4 John Kersey, Algebra (London, 1673), Book IV, p. 177. 6 Edward Hatton, An Intire System of Arithmetic (London, 1721), p. 287. 6 Edmund Stone, New Mathematical Dictionary (London, 1726; 2d ed., 1743), art. "Character." 7 Jarncs Hodgson, A System of Mathematics, Vol. I (London, 1723), p. 10. 8 Encyclopedic ou Dictionnaire raissonne, etc. (Diderot), Vol. VI (Lausanne et Berne, 1781), art. "Caractere." 9 H. S. Hall and F. H. Stevens, Euclid's Elements, Parts I and U (London, 1889), p. 10. 10 Theodor Reye, Die Geometric der Laqe (5th ed.; Leipzig, 1909), 1. Abteilung, p. 83. 11 John Caswell, "Doctrine of Trigonometry," in Wallis' Algebra (1685). 12 John Caswell, "Trigonometry," in ibid. 18 Francisci Dulaurens, Specimina mathematica duobus libris comprehensa (Paris, 1667), "Symbols." 404 A HISTORY OF MATHEMATICAL NOTATIONS Jones, 1 Emerson, 2 Hutton, 3 Fuss, 4 Steenstra, 6 Kliigel, 6 Playfair, 7 Kambly, 8 Wentworth, 9 Fiedler, 10 Casey, 11 Lieber and von Ltihmann, 12 Byerly, 13 Muller, 14 Mehler, 15 C, Smith, 16 Beman and Smith, 17 Layng, 18 Hopkins, 19 Robbins, 20 the National Committee (in the U.S.A.). 21 The plural "angles" is designated by Caswell Z Z ; by many others thus, A. Caswell also writes ZZ.Z. for the "sum of two angles," and X for the "difference of two angles." From these quotations it is evident that the sign Z for angle enjoyed wide popularity in different countries. However, it had rivals. 361. Sometimes the same sign is inverted, thus 7 as in John Ward. 22 Sometimes it is placed so as to appear A, as in the Ladies 1 William Jones, Synopsis palmariorum matheseos (London, 1706), p. 221. 2 [W. Emerson], Elements of Geometry (London, 1763), p. 4. 8 Charles Hutton, Mathematical and Philosophical Dictionary (1695), art. "Characters.' 1 4 Nicolas Fuss, Lemons de geometrie (St. Petersbourg, 1798), p. 38. 5 Pibo Steenstra, Grondbeginsels der Meetkunst (Leyden, 1779), p. 101. 6 G. S. Kliigel, Math. Worterbuch, fortgesetzt von C. B. Mollweide und J. A. Grunert, 5. Theil (Leipzig, 1831), art. "Zeichen." 1 John Playfair, Elements of Geometry (Philadelphia, 1855), p. 114. 8 L. Kambly, Die Elementar-Mathematik, 2. Theil: Planimetrie, 43. Aufl. (Breslau, 1876). 9 G. A. Wentworth, Ekments of Geometry (Boston, 1881; Preface, 1878). 10 W. Fiedler, Darstellende Geometric, 1. Theil (Leipzig, 1883), p. 7. 11 John Casey, Sequel to the First Six Books of the Elements of Euclid (Dublin, 1886). 13 H. Lieber und F. von Liihmann, Geometrische Konstruktions-Aufgaben, 8. Aufl. (Berlin, 1887), p. 1. 13 W. E. Byerly's edition of Chauvenet's Geometry (Philadelphia, 1905), p. 44. 14 G. Muller, Zeichnende Geometric (Esslingen, 1889), p. 12. 15 F. G. Mehler, Hauptsdtze der Elementar Mathematik, 8. Aufl. (Berlin, 1894), p. 4. 16 Charles Smith, Geometrical Conies (London, 1894). 17 W. W. Beman and D. E. Smith, Plane and Solid Geometry (Boston, 1896), p. 10. 18 A. E. Layng, Euclid's Elements of Geometry (London, 1890), p. 4. 19 G. Irving Hopkins, Inductive Plane Geometry (Boston, 1902), p. 12. 20 E. R. Robbins, Plane and Solid Geometry (New York, [1906]), p. 16. 21 Report by the National Committee on Mathematical Requirements, under the auspices of the Mathematical Association of America, Inc. (1923), p. 77. 22 John Ward, The Young Mathematiciana' Guide (9th ed.; London, 1752), p. 301, 369. GEOMETRY 405 Diary 1 and in the writings of Reyer, 2 Bolyai, 8 and Ottoni. 4 This posi- tion is widely used in connection with one or three letters marking an angle. Thus, the angle ABC is marked by L. N. M. Carnot 5 ABC in his Geometric de position (1803); in the Penny Cyclopedia( 1839), arti- cle "Sign," there is given A B; Binet, 6 Mobius, 7 and Favaro 8 wrote ab as the angle formed by two straight lines a and b; Favaro wrote also PDC. The notation a b is used by Stob and Gmeiner, 9 so that /\ /\ X\ /\ X\ a b=b a; Nixon 10 adopted A, also ABC; the designation APM is found in Enriques, 11 Borel, 12 and Durrell. 13 362. Some authors, especially German, adopted the sign < for angle. It is used by Spitz, 14 Fiedler, 15 Halsted, 16 Milinowski, 17 Meyer, 13 1 Leybourne's Ladies Diary, Vol. IV, p. 273. 2 Samuel Reyhers .... Euclides, dessen VI. erste Backer auf sonderbare Art mil algebraischen Zeichen, also eingerichtet, sind, dass man derselben Beweise auch in anderen Sprachen gebrauchen kann (Kiel, 1698). 8 Wolfgang! Bolyai de Bolya, Tentamen (2d ed.), Tome II (Budapestini, 1904; 1st ed., 1832), p. 361. 4 C. B. Ottoni, Elementos de Geometria e Trigonometria (4th ed.; Rio de Janeiro, 1874), p. 67. 5 See Ch. Babbage, "On the Influence of Signs in Mathematical Reasoning," Transactions Cambridge Philos. Society, Vol. II (1827), p. 372. J. P. Binet in Journal de Vecole polyt., Vol. IX, Cahier 16 (Paris, 1813), p. 303. 7 A. F. Mobius, Gesammelte Werke, Vol. I (Leipzig, 1885), "Barycyentrischer Calcul, 1827," p. 618. 8 A. Favaro, Lemons de Statique graphique, trad, par Paul Terrier, 1. Partie (Paris, 1879), p. 51, 75. 9 O. Stolz und J. A. Gmeiner, Theoretische Arithmetik (Leipzig), Vol. II (1902), p. 329, 330. 10 R. C. J. Nixon, Euclid Revised (3d ed.; Oxford, 1899), p. 9. 11 Fedcrigo Enriques, Questioni riguardanti la geometria elementare (Bologna, 1900), p. 67. 12 Emile Borel, Algebre (2d cycle; Paris, 1913), p. 367. 13 Clement V. Durell, Modern Geometry; The Straight Line and Circle (London, 1920), p. 7, 21, etc. 14 Carl Spitz, Lehrbuch der ebenen Geometric (Leipzig und Heidelberg, 1862), p. 11. 15 W. Fiedler, Darstellende Geometric, 1. Theil (Leipzig, 1883), p. 7. 16 George Bruce Halsted, Mensuration (Boston, 1881), p. 28; Elementary Syn- thetic Geometry (New York, 1892), p. vii. 17 A. Milinowski, Elcm.-Synth. Geom. der Kegelschnitte (Leipzig, 1883), p. 3. 18 Fricdrich Meyer, Dritter Cursus der Planimetrie (Halle, a/S, 1885), p. 81. 406 A HISTORY OF MATHEMATICAL NOTATIONS Fialkowski, 1 Henrici and Treutlein, 2 Bruckner , 3 Doehlerfiann, 4 Schur, 5 Bernhard, 6 Auerbach and Walsh, 7 Mangoldt. 8 If our quotations arc representative, then this notation for angle finds its adherents in Germany and the United States. A slight modi- fication of this sign is found in Byrne 9 <0. Among sporadic representations of angles are the following: The capital letter 10 L, the capital letter 11 V, or that letter inverted, 12 A, the inverted capital letter 13 \/J the perpendicular lines 14 _] or L, pq the angle made by the lines 15 p and #, (a6) the angle between the rays, 16 a and b, ab the angle between the lines 17 a and b, or (u, v) the angle 18 formed by u and v. 363. Passing now to the designation of special angles we find ^ used to designate an oblique angle. 19 The use of a pictograph for the designation of right angles was more frequent in former years than now and occurred mainly in English texts. The two perpendicular lines L: to designate "right angle" are found in Reyher; 20 he lets 1 N. Fialkowski, Praklische Geometric (Wien, 1892), p. 15. 2 J. Henrici und P. Treutlein, Lehrbuch der Elementar-Geometrie, 1. Teil, 3. Aufl. (Leipzig, 1897), p. 11. 3 Max Bruckner, Vielecke und Vielfache-Theorie und Geschichte (Leipzig, 1900), p. 125. 4 Karl Doehlemann, Projektive Geometric, 3. Aufl. (Leipzig, 1905), p. 133. 5 F. Schur, Grundlagen der Geometric (Leipzig und Berlin, 1909), p. 79. 9 Max Bernhard, Darstellende Geometric (Stuttgart, 1909), p. 267. 7 Matilda Auerbach and Charles B. Walsh, Plane Geometry (Philadelphia, [1920]), p. vii. 8 Hans V. Mangoldt, Einfiihrung in die hohere Mathematik, Vol. I (Leipzig, 1923), p. 190. 9 Oliver Byrne, Elements of Euclid (London, 1847), p. xxviii. 10 John Wilson, Trigonometry (Edinburgh, 1714), " Characters Explained." 11 A. Saverien, Dictionnaire de math, et phys. (Paris, 1753), "Caractere." 12 W. Bolyai, Tentamen (2d ed.), Vol. I (1897), p. xi. 13 Joseph Fenn, Euclid (Dublin, 1769), p. 12; J. D. Blassiere, Principes de geometric elementaire (The Hague, 1782), p. 16. 14 H. N. Robinson, Geometry (New York, 1860), p. 18; ibid. (15th ed., New York), p. 14. 16 Charlotte Angas Scott, Modern Analytical Geometry (London, 1894), p. 253. 16 Heinrich Schroter, Theorie der Kegelschnitte (2d ed; Leipzig, 1876), p. 5. 17 J. L. S. Hatton, Principles of Projective Geometry (Cambridge, 1913), p. 9. 18 G. Peano, Formulaire mathematique (Turin, 1903), p. 266. 19 W. N. Bush and John B. Clarke, Elements of Geometry (New York, [1905]). 20 Samuel Reyhers Euclides (Kiel, 1698). GEOMETRY 407 MIL stand for "angle A is a right angle," a symbolism which could be employed in any language. The vertical bar stands for equality ( 263). The same idea is involved in the signs a t 6, i.e., "angle a is equal to angle 6." The sign L for right angle is found in Jones, 1 Hatton, 2 Savericn, 3 Fenn, 4 and Steenstra. 5 Kersey 6 uses the sign -i. , Byrne 7 Q>. Mach 8 marks right angles i. The Frenchman Hdrigone 9 used the sign J, the Englishman Dupius 10 "1 for right angle. James Mills Peirce, 11 in an article on the notation of angles, uses 11 Greek letters to denote the directions of lines, without reference to their length. Thus if p denotes the axis in a system of polar co-ordi- nates, the polar angle will be p." Accordingly, = # More common among more recent American and some English writers is the designation "rt. Z" for right angle. It is found in G. A. Wentworth, 12 Byerly's Chauvenet, Hall and Stevens, 14 Beman and Smith, 15 Hopkins, 16 Robbins, 17 and others. Some writers use instead of pictographs of angles abbreviations of the word. Thus Legendre 18 sometimes writes "Angl. ACB"\ 1 William Jones, Synopsis palrnariorum matheseos (London, 1706), p. 221. 2 Edward Hatton, An Intire System of Arithmetik (London, 1721), p. 287. 3 A. Saverien, Dictionnaire, "Caractere." 4 Joseph Fenn, Euclid (Dublin, 1769), p. 12. 5 Pibo Steenstra, Grondbeginsels der Meetkunst (Leyden, 1779), p. 101. 6 John Kersey, Algebra (London, 1673), Book IV, p. 177. 7 Oliver Byrne, The Elements of Euclid (London, 1847), p. xxviii. 8 E. Mach, Space and Geometry (trans. T. J. McCormack, 1906), p. 122. 9 P. Herigone, Cursus mathematicus (Paris, 1634), Vol. I, "Explicatio no- tarum." 10 N. F. Dupius, Elementary Synthetic Geometry (London, 1889), p. 19. 11 J. D. Runkle's Mathematical Monthly, Vol. I, No. 5 (February, 1859), p. 168, 169. 12 G. A. Wentworth, Elements of Plane and Solid Geometry (3d ed. ; Boston, 1882), p. 14. 13 W. E. Byerly's edition of ChauveneCs Geometry (1887). 14 H. S. Hall and F. II. Stevens, Euclid 's Elements, Parts I and II (London, 1889), p. It). 15 W. W. Beman and D. E. Smith, Plane and Solid Geometry (Boston, 1896), p. 10. 16 G. I. Hopkins, Inductive Plane Geometry (Boston, 1902), p. 12. 17 E. R. Robbins, Plane and Solid Geometry (New York, [1906]), p. 16. 18 A. M. Legendre, Elements de Geomelrie (Paris, 1794), p. 42. 408 A HISTORY OF MATHEMATICAL NOTATIONS A. von Frank, 1 "Wkl," the abbreviation for Winkel, as in "Wkl DOQ." The advent of non-Euclidean geometry brought Lobachevski's notation n (p) for angle of parallelism. 3 The sign -Y- to signify equality of the angles, and _L to signify the equality of the sides of a figure, are mentioned in the article "Caractere" by D'Alembert in Diderot* Encyclopedic of 1754 and of 1781 s and in the Italian translation of the mathematical part (1800) ; also in Rees's Cyclopaedia (London, 1819), article "Characters," and in E. Stone's New Mathematical Dictionary (London, 1726), article "Characters," but Stone defines _Y~ as signifying "equiangular or similar." The symbol is given also by a Spanish writer as signifying angulos iguales.* The sign to signify "equal number of degrees" is found in Palmer and Taylor's Geometry* but failed to be recom- mended as a desirable symbol in elementary geometry by the Na- tional Committee on Mathematical Requirements (1923), in their Report, page 79. Halsted suggested the sign for spherical angle and also the letter 12 to represent a "steregon," the unit of solid angle. 6 364. Signs for "perpendicular." The ordinary sign to indicate that one line is perpendicular to another, _L, is given by H6rigone 7 in 1634 and 1644. Another Frenchman, Dulaurens, 8 used it in 1667. In 1673 Kersey 9 in England employed it. The inverted capital letter j, was used for this purpose by Caswell, 10 Jones, 11 Wilson, 12 Saverien, 18 1 A. von Frank in Archiv der Mathematik und Physik von J, A. Grunert (2d ser.), Vol. XI (Leipzig, 1892), p. 198. 2 George Bruce Halsted, N. Lobatschewsky, Theory of Parallels (Austin, 1891), p. 13. 3 Encydopedie au Dictionnaire raisonne des sciences , ... by Diderot, Vol. VI (Lausanne et Berne, 1781), art. "Caractere." 4 Antonio Serra y Oliveres, Manual de la Tipografia Espanola (Madrid, 1852), p. 70. 6 C. I. Palmer and D. P. Taylor, Plane Geometry (1915), p. 16. 6 G. B. Halsted, Mensuration (Boston, 1881), p. 28. 7 Pierre Herigone, Curvus mrthematicus, Vol. I (Paris, 1634), "Explicatio notarum." 8 F. Dulaurens, Specimina mathematica dudbus libris comprehewa (Paris, 1667), "Symbols. 9 John Kersey, Algebra (London, 1673), Book IV, p. 177. 10 J. Caswell's Trigonometry in J. Wallifl' Algebra (1685). 11 W. Jones, op. tit., p. 253. 12 J. Wilson, Trigonometry (Edinburgh, 1714), "Characters Explained." 13 A. Saverien, Lhctionnaire, "Caractere." GEOMETRY 409 and Mauduit. 1 Emerson 2 has the vertical bar extremely short, -*-. In the nineteenth century the symbol was adopted by all writers using pictographs in geometry. Sometimes _k was used for ' 'perpendiculars. ' ' Thomas Baker 3 adopted the symbol ^ for perpendicular. 365. Signs for triangle, square, rectangle, parallelogram. The signs A, D, CH or [), O are among the most widely used pictographs. We have already referred to their occurrence down to the time of Hdrigone and Oughtred (184). The O for parallelogram is of rare occurrence in geometries preceding the last quarter of the nineteenth century, while the A, D, and CD occur in van Schooten, 4 Dulaurens, 5 Kersey, 6 Jones, 7 and Saverien. 8 Some authors use only two of the three. A rather curious occurrence is the Hebrew letter "mem," C2, to repre- sent a rectangle; it is found in van Schooten, 9 Jones, 10 John Alexander, 11 John I Bernoulli, 12 Ronayne, 13 Kliigel's Worterbuch and De Graaf. 15 Newton, 16 in an early manuscript tract on fluxions (October, 1666), indicates the area or fluent of a curve by prefixing a rectangle to the "~"* / ordinate ( 622), thus D r-r- , where x is the abscissa, and the ao~iXx fraction is the ordinate. After about 1880 American and English school geometries came to employ less frequently the sign a for rectangle and to introduce more often the sign O for parallelogram. Among such authors are 1 A. R. Mauduit, Inleiding tot de Kleegel-Sneeden (The Hague, 1763), "Sym- bols." 2 [W. Emerson], Elements of Geometry (London, 1763). 3 Thomas Baker, Geometrical Key (London, 1684), list of symbols. 4 Fr. van Schooten, Exercitationvm mathematicorvm liber primm (Leyden, 1657). 6 F. Dulaurens, loc. tit., "Symbols." 6 J. Kersey, Algebra (1673). 7 W. Jones, op. til., p. 225, 238. 8 Saverien, loc. tit. 9 Franciscus van Schooten, op. cit. (Leyden, 1657), p. 67. 10 W. Jones, op. til., p. 253. 11 Synopsis Algebraica, opus poslhumum lohannis Alexandri (London, 1693), p. 67. 12 John Bernoulli in Ada eruditorum (1689), p. 586; ibid. (1692), p. 31. 18 Philip Ronayne, Treatise of Algebra (London, 1727), p. 3. M J. G. Klugel, Math. Worterbuch, 5, Theil (Leipzig, 1831), "Zeichen." 16 Abraham de Graaf, Algebra of Stelkonst (Amsterdam, 1672), p. 81. 18 S. P. Rigaud, Historical Essay on .... Newton's Printipia (Oxford, 1838), Appendix, p. 23. 410 A HISTORY OF MATHEMATICAL NOTATIONS Halsted, 1 Wentworth, 2 Byerly, 8 in his edition of Chauvenet, Beman and Smith, 4 Layng, 5 Nixon, 6 Hopkins, 7 Robbins, 8 and Lyman. 9 Only seldom do both O and 7 appear in the same text. Halsted 10 denotes a parallelogram by \\g'm. Special symbols for right and oblique spherical triangles, as used by Jean Bernoulli in trigonometry, are given in Volume II, 524. 366. The square as an operator. The use of the sign D to mark the operation of squaring has a long history, but never became popu- lar. Thus N. Tartaglia 11 in 1560 denotes the square on a line tc in the expression "il D de. tc." Cataldi 12 uses a black square to indicate the square of a number. Thus, he speaks of 8J-J-, "il suo | e 75 \\ -fj}." Stampioen 13 in 1640 likewise marks the square on BC by the "D BC." Caramvel 14 writes "D25. est Quadratum Numeri 25. hocest, 625." A. de Graaf 15 in 1672 indicates the square of a binomial thus: I/a |/6, "zijn D is a+b2i/ab." Johann I Bernoulli 16 wrote 3 D Vax+xx for 3|/(ax+x 2 ) 2 . Jakob Bernoulli in 1690 17 designated 1 G. B. Halsted, Elem. Treatise on Mensuration (Boston, 1881), p. 28. 2 G. A. Wentworth, Elements of Plane and Solid Geometry (3d ed. ; Boston, 1882), p. 14 (1st ed., 1878). 3 W. E. Byerly 's edition of Chauvenet' s Geometry (1887), p. 44. 4 W. W. Beman and D. E. Smith, Plane and Solid Geometry (Boston, 1896), p. 10. 5 A. E. Layng, Euclid's Elements of Geometry (London, 1890), p. 4. 8 R. C. J. Nixon, Euclid Revised (3d ed., Oxford, 1899), p. 6. 7 G. J. Hopkins, Inductive Plane Geometry (Boston, 1902), p. 12. 8 E. R. Robbins, Plane and Solid Geometry (New York, [1906]), p. 16. 9 E. A. Lyman, Plane and Solid Geometry (New York, 1908), p. 18. 10 G. B. Halsted, Rational Geometry (New York, 1904), p. viii. 11 N. Tartaglia, La Quinta parte del general trattato de nvmeri et misvre (Venice, 1560), fols.82A and 83A. 12 Trattato del Modo Brevissimo de trouare la Radice quadra delli numeri, .... Di Pietro Antonio Cataldi (Bologna, 1613), p. 111. 13 J. Stampioen, Wis-Konstich ende Reden-maetich Bewys ('S Graven-Hage, 1640), p. 42. 14 Joannis Caramvelis mathesis biceps, vetus, et nova (1670), p. 131. 15 Abraham de Graaf, Algebra of Stelkonst (Amsterdam, 1672), p. 32. 16 Johannis I Bernoulli, Lectiones de calculo differentialium .... von Paul Schafheitlin, Separatabdruck aus den Verhandlungen der Naturforschenden Ge- sellschaft in Basel, Vol. XXXIV (1922). 17 Jakob Bernoulli in Ada erudiiorum (1690), p. 223. GEOMETRY 411 the square of by QOf , but in his collected writings 1 it is given in the modern form (|) 2 . Sometimes a rectangle, or the Hebrew letter "mem," is used to signify the product of two polynomials. 2 367. Sign for circle. Although a small image of a circle to take the place of the word was used in Greek time by Heron and Pappus, the introduction of the symbol was slow. H6rigone used O, but Oughtred did not. One finds O in John Kersey, 3 John Caswell, 4 John Ward, 5 P. Steenstra, 6 J. D. Blasstere, 7 W. Bolyai, 8 and in the writers of the last half -century who introduced the sign O for paral- lelogram. Occasionally the central dot is omitted and the symbol O is used, as in the writings of Reyher 9 and Saverien. Others, Fenn for instance, give both O and O, the first to signify circumference, the second circle (area). Caswell 10 indicates the perimeter by O. Metius 11 in 1623 draws the circle and a horizontal diameter to signify circulus. 368. Signs for parallel lines. Signs for parallel lines were used by Heron and Pappus ( 701); Hrigone used horizontal lines = ( 189) as did also Dulaurens 12 and Reyher, 13 but when Recorders sign of equality won its way upon the Continent, vertical lines came to be used for parallelism. We find || for "parallel" in Kersey, 14 Caswell, Jones, 15 Wilson, 16 Emerson, 17 Kambly, 18 and the writers of the last 1 Opera Jakob Bernoulli.?, Vol. I, p. 430, 431; see G. Enestrom, Bibliotheca mathematica (3d ser.), Vol. IX (1908-9), p. 207. 2 See P. Herigone, Cursus mathematici (Paris, 1644), Vol. VI, p. 49. 3 John Kersey, Algebra (London, 1673), Book IV, p. 177. 4 John Caswell in Wallis' Treatise of Algebra, "Additions and Emendations/' p. 166. For "circumference" Caswell used the small letter c. 5 J. Ward, The Young Mathematician's Guide (9th cd.; London, 1752), p. 301, 369. 6 P. Steenstra, Grondbeginsels der Meetkunst (Ley den, 1779), p. 281. 7 J. D. Blassiere, Principes de geometric 6lemcntaire (The Hague, 1723), p. 16. 8 W. Bolyai, Tentamen (2d ed.), Vol. II (1904), p. 361 (1st ed., 1832). 9 Samuel Reyher s, Euclides (Kiel, 1698), list of symbols. 10 John Caswell in WalhV Treatise of Algebra (1685), "Additions and Emenda- tions," p. 166. 11 Adriano Metio, Praxis nova geometrica (1623), p. 44. 12 Fr. Dulaurens, Spedmina mathematica (Paris, 1667), "Symbols." 13 S. Reyher, op. cit. (1698), list of symbols. 14 John Kersey, Algebra (London, 1673), Book IV, p. 177. 16 W. Jones, Synopsis palmariorum matheseos (London, 1706). 16 John Wilson, Trigonometry (Edinburgh, 1714), characters explained. 17 [W. Emerson], Elements of Geometry (London, 1763), p. 4. 18 L. Kambly, Die Elementar-Mathematik, 2. Theil, Planimetrie, 43. Aufl. (Breslau, 1876), p. 8. 412 A HISTORY OF MATHEMATICAL NOTATIONS fifty years who have been already quoted in connection with other pictographs. Before about 1875 it does not occur as often as do A, D, a. Hall and Stevens 1 use "par 1 or ||" for parallel. Kambly 2 men- tions also the symbols -ff and 4= for parallel. A few other symbols are found to designate parallel. Thus John Bolyai in his Science Absolute of Space used |||. Karsten 3 used 4; he says: "Man pflege wohl das Zeichen :$ statt des Worts: Parallel der Klirze wegen zu gebrauchen." This use of that symbol occurs also in N. Fuss. 4 Thomas Baker 5 employed the sign =* . With Kambly # signifies rectangle. Haseler 6 employs $ as "the sign of parallelism of two lines or surfaces." 369. Sign for equal and parallel. #is employed to indicate that two lines are equal and parallel in Klugel's Worterbuch; 1 it is used by H. G. Grassmann, 8 Lorey, 9 Fiedler, 10 Henrici and Treutlein. 11 370. Signs for arcs of circles. As early a writer as Plato of Tivoli ( 359) used ab to mark the arc ab of a circle. Ever since that time it has occurred in geometric books, without being generally adopted. It is found in Hrigone, 12 in Reyher, 13 in Kambly, 14 in Lieber and Luhmann. 16 W. R. Hamilton 18 designated by ^LF the arc "from F to L." These 1 H. S. Hall and F. H. Stevens, Euclid's Elements, Parts I and II (London, 1889), p. 10. 2 L. Kambly, op. Git., 2. Theil, Planimetrie, 43. Aufl. (Breslau, 1876), p. 8. 3 W. J. G. Karsten, Lehrbegrif der gesamten Mathematik, 1. Theil (Greifswald, 1767), p. 254. 4 Nicolas Fuss, Lemons de geometric (St. Petersbourg, 1798), p. 13. 5 Thomas Baker, Geometrical Key (London, 1684), list of symbols. J. F. Haseler, Anfangsgrunde der Arith., Alg., Geom. und Trig. (Lemgo), Elementar-Geometrie (1777), p. 72. 7 G. S. Kliigel, Mathemalisches Worterbuch, fortgesetzt von C. B. Mollweide, J. A. Grunert, 5. Theil (Leipzig, 1831), "Zeichen." 8 H. G. Grassmann, Ausdehnungslehre von 1844 (Leipzig, 1878), p. 37; Werke by F. Engel (Leipzig, 1894), p. 67. 9 Adolf Lorey, Lehrbuch der ebenen Geomelrie (Gera und Leipzig, 1868), p. 52. 10 Wilhelm Fiedler, Darstellende Geometric, 1. Theil (Leipzig, 1883), p. 11. 11 J. Henrici und P. Treutlein, Lehrbuch der Elementar-Geometrie, 1. Teil, 3. Aufl. (Leipzig, 1897), p. 37. 12 P. Herigone, op. ciL (Paris, 1644), Vol. I, "Explicatio notarum." 13 Samuel Reyhers, Euclides (Kiel, 1698), Vorrede. 14 L. Kambly, op. tit. (1876). 16 H. Lieber und F. von Luhmann, Geomelrische Konstructions-Aufgaben, 8. Aufl. (Berlin, 1887), p. 1. 16 W. R. Hamilton in Cambridge & Dublin Math'l. Journal, Vol. I (1846), p. 262. GEOMETRY 413 references indicate the use of ^ to designate arc in different countries. In more recent years it has enjoyed some popularity in the United States, as is shown by its use by the following authors: Halsted, 1 Wells, 2 Nichols, 3 Hart and Feldman, 4 and Smith. 5 The National Com- mittee on Mathematical Requirements, in its Report (1923), page 78, is of the opinion that "the value of the symbol ^ in place of the short word arc is doubtful/' In 1755 John Landen 6 used the sign (PQR) for the circular arc which measures the angle PQR, the radius being unity. 371. Other pictographs. We have already referred to Herigone's use ( 189) of 5< and 6< to represent pentagons and hexagons. Reyher actually draws a pentagon. Occasionally one finds a half- circle and a diameter to designate a segment, and a half-circle with- out marking its center or drawing its diameter to designate an arc. Reyher in his Euclid draws / \ for trapezoid. Pictographs of solids are very rare. We have mentioned ( 359) those of Metius. Saverein 7 draws |? A> MM > HI to stand, respec- tively, for cube, pyramid, parallelepiped, rectangular parallelopiped, but these signs hardly belong to the category of pictographs. Dulaurens 8 wrote IH for cube and GO for aequi quadrimensum. Joseph Fenn 9 draws a small figure of a parallelopiped to represent that solid, as Metius had done. Halsted 10 denotes symmetry by -I- . Some authors of elementary geometries have used algebraic sym- bols and no pictographs (for instance, Isaac Barrow, Karsten, Tac- quet, Leslie, Legendre, Playfair, Chauvenet, B. Peirce, Todhunter), but no author since the invention of symbolic algebra uses pictographs without at the same time availing himself of algebraic characters. 372. Signs for similarity and congruence. The designation of "similar,'' "congruent/' "equivalent," has brought great diversity of notation, and uniformity is not yet in sight. Symbols for similarity and congruence were invented by Leibniz. 1 G. B. Halsted, Mensuration (Boston, 1881). 2 Webster Wells, Elementary Geometry (Boston, 1886), p. 4. 3 E. H. Nichols, Elements of Constructional Geometry (New York, 1896). 4 C. A. Hart and D. D. Feldman, Plane Geometry (New York, [1911]), p. viii. 6 Eugene R. Smith, Plane Geometry (New York, 1909), p. 14. 6 John Landen, Mathematical Lucubrations (London, 17.55), Sec. Ill, p. 93. 7 A. Saverein, Dictionnaire, "Caractere." 8 F. Dulaurens, op. cit. (Paris, 1667), "Symbols." 9 Joseph Fenn, Euclid's Elements of Geometry (Dublin, [ca. 1769]), p. 319. 10 G. B. Halsted, Rational Geometry (New York, 1904), p. viii. 414 A HISTORY OF MATHEMATICAL NOTATIONS In Volume II, 545, are cited symbols for "coincident" and "congru- ent" which occur in manuscripts of 1679 and were later abandoned by Leibniz. In the manuscript of his Characteristica Geometrica which was not published by him, he says: "similitudinemitanotabimus: a~b.' n The sign is the letter S (first letter in similis) placed horizontally. Having no facsimile of the manuscript, we are dependent upon the editor of Leibniz' manuscripts for the information that the sign in question was ~ and not ^>. As the editor, C. I. Gerhardt, inter- changed the two forms (as pointed out below) on another occasion, we do not feel certain that the reproduction is accurate in the present case. According to Gerhardt, Leibniz wrote in another manuscript r^ for congruent. Leibniz 7 own words are reported as follows: "ABC ~.CDA. Nam ~ inihi est signum similitudinis, et = aequalitatis, unde congruentiae signum compono, quia quae simul et similia et aequalia sunt, ea congrua sunt." 2 In a third manuscript Leibniz wrote |r^| for coincidence. An anonymous article printed in the Miscellanea Berolinensia (Berlin, 1710), under the heading of "Monitum de characteribus alge- braicis," page 159, attributed to Leibniz and reprinted in his col- lected mathematical works, describes the symbols of Leibniz; ^ for similar and ^2. for congruent ( 198). Note the change in form; in the manuscript of 1679 Leibniz is reported to have adopted the form ~, in the printed article of 1710 the form given is ^. Both forms have persisted in mathematical writings down to the present day. As re- gards the editor Gerhardt, the disconcerting fact is that in 1863 he reproduces the ^ of 1710 in the form 3 ^. The Leibnizian symbol ~ was early adopted by Christian von Wolf; in 1716 he gave ~ for Aehnlichkeit, 4 and in 1717 he wrote " = et ~" for "equal and similar." 5 These publications of Wolf are the earliest in which the sign ^ appears in print. In the eighteenth and early part of the nineteenth century, the Leibnizian symbols for "similar" and "congruent" were seldom used in Europe and not at all in England and America. In England ^ or ^ usually expressed "difference," as defined by Oughtred. In the eighteenth century the signs for congruence occur much less frequently even than the signs 1 Printed in Leibnizens Math. Schrifien (ed. C. I. Gerhardt), Vol. V, p. 153. 2 Op. tit., p. 172. 3 Leibnizens Math. Schriften, Vol. VII (1863), p. 222. 4 Chr. Wolffen, Math. Lexicon (Leipzig, 1716), "Signa." 5 Chr. V. Wolff, Elementa Matheseos universalis (Halle, 1717), Vol. I, 236; see Tropfke, op. tit., Vol. IV (2d ed., 1923), p. 20. GEOMETRY 415 >r similar. We have seen that Leibniz' signs for congruence did not se both lines occurring in the sign of equality = . Wolf was the first to se explicitly ~ and = for congruence, but he did not combine the NO into one symbolism. That combination appears in texts of the itter part of the eighteenth century. While the ~ was more involved, nee it contained one more line than the Leibnizian ^, it had the dvantage of conveying more specifically the idea of congruence as le superposition of the ideas expressed by ~ and =. The sign ^> >r "similar" occurs in Camus' geometry, 1 ^ for "similar" in A. R. lauduit's conic sections 2 and in Karsten, 3 ^ in Blassiere's geometry, 4 = for congruence in Haseler's 5 and Reinhold's geometries, 6 ^ for milar in Diderot's Encyclopedic? and in Lorenz' geometry. 8 In ItigeFs Worterbuch? one reads, "^ with English and French authors icans difference" ; "with German authors ^ is the sign of similarity" ; Leibniz and Wolf have first used it." The signs ~ and ^ are used y Mollweide; 10 ~ by Steiner 11 and Koppe; 12 ^ is used by Prestel, 13 = by Spitz; 14 ~ and = are found in Lorey's geometry, 15 Kambly's 1 C. E. L. Camus, Siemens de geometrie (nouvelle 6d.; Paris, 1755). 2 A. R. Mauduit, op. tit. (The Hague, 1763), "Symbols." 3 W. J. G. Karsten, Lehrbegrif der gesamten Mathematik, 1. Theil (1767), 348. 4 J. D. BlassieTe, Printipes de geometrie ttementaire (The Hague, 1787), 16. 8 J. F. Ilaseler, op. tit. (Lemgo, 1777), p. 37. 6 C. L. Reinhold, Arithmetica Forensis, 1. Theil (Ossnabriick, 1785), p. 361. 7 Diderot Encyclopedic on Dictionnaire raisone des sciences (1781; 1st ed., 54), art. "Caractere" by D'Alembert. See also the Italian translation of the athematical part of Diderot's Encyclopedic, the IHzionario enticlopedico delle tiematiche (Padova, 1800), "Carattcrc." 8 J. F. Lorenz, Grundriss der Arithmetik und Geometrie (HelmstMt, 1798), p. 9. 9 G. S. Kliigel, Mathematisches Worterbuch, fortgesetzt von C. B. Mollweide, A. Grunert, 5. Theil (Leipzig, 1831), art. "Zeichen." 10 Carl B. Mollweide, Euklid's Elemente (Halle, 1824). 11 Jacob Steiner, Geometrische Constructionen (1833); Ostwald's Klassiker, No. ,p.6. 12 Karl Koppe, Planimelrie . (Essen, 1852), p. 27. u M. A. F. Prestel, Tabelarischer Grundriss der Experimental-physik (Emden, 56), No. 7. 14 Carl Spitz, Lehrbuch der ebenen Geometrie (Leipzig und Heidelberg, 1862), 41. 15 Adolf Lorey, Lehrbuch der ebenen Geometrie (Gera und Leipzig, 1868), 118. 416 A HISTORY OF MATHEMATICAL NOTATIONS Planimetrie^ and texts by Frischauf 2 and Max Simon. 8 Lorey's book contains also the sign ^ a few times. Peano 4 uses ^ for "similar" also in an arithmetical sense for classes. Perhaps the earliest use of ^ and ^ for "similar" and "congruent" in the United States are by G. A. Hill 5 and Halsted. 6 The sign ^ for "similar" is adopted by Henrici and Treutlein, 7 ^ by Fiedler, 8 ^ by Fialkowski, 9 ^ by Beman and Smith. 10 In the twentieth century the signs entered geometries in the United States with a rush: ^ for "congruent" were used by Busch and Clarke; 11 ^ by Meyers, 12 ^ by Slaught and Lennes, 13 ^ by Hart and Feldman; 14 ^ by Shutts/ 6 E. R. Smith, 16 Wells and Hart, 17 Long and Brenke; 18 ^ by Auerbach and Walsh. 19 That symbols often experience difficulty in crossing geographic or national boundaries is strikingly illustrated in the signs ~ and ^. The signs never acquired a foothold in Great Britain. To be sure, the symbol *-s was adopted at one time by a member of the University 1 L. Kambly, Die Elementar-Mathematik, 2. Thcil, Planimetrie, 43. Aufl. (Breslau, 1876). 2 J. Frischauf, Absolute Geometric (Leipzig, 1876), p. 3. 3 Max Simon, Euclid (1901), p. 45. 4 G. Pcano, Formulaire de mathematiques (Turin, 1894), p. 135. 6 George A. Hill, Geometry for Beginners (Boston, 1880), p. 92, 177. 6 George Bruce Halsted, Mensuration (Boston, 1881), p. 28, 83. 7 J. Henrici und P. Treutlein, Elementar-Geometrie (Leipzig, 1882), p. 13, 40. 8 W. Fiedler, Darstellende Geometric, 1. Theil (Leipzig, 1883), p. 60. 9 N. Fialkowski, Praktische Geometrie (Wien, 1892), p. 15. 10 W. W. Bernan and D. E. Smith, Plane and Solid Geometry (Boston, 1896), p. 20. 11 W. N. Busch and John B. Clarke, Elements of Geometry (New York, 1905]). 12 G. W. Meyers, Second-Year Mathematics for Secondary Schools (Chicago, 1910), p. 10. 13 II. E. Slaught and N. J. Lennes, Plane Geometry (Boston, 1910). 14 C. A. Hart and D. D. Feldman, Plane Geometry (New York, 1911), p. viii. 16 G. C. Shutts, Plane and Solid Geometry [1912], p. 13. 16 Eugene R. Smith, Solid Geometry (New York, 1913). 17 W. Wells and W. W. Hart, Plane and Solid Geometry (Boston, [1915]), p. x. 18 Edith Long and W. C. Brenke, Plane Geometry (New York, 1916), p. viii. 19 Matilda Auerbach and Charles Burton Walsh, Plane Geometry (Philadelphia, [1920]), p. xi. GEOMETRY 417 of Cambridge, 1 to express "is similar to" in an edition of Euclid. The book was set up in type, but later the sign was eliminated from all parts, except one. In a footnote the student is told that "in writing out the propositions in the Senate House, Cambridge, it will be ad- visable not to make use of this symbol, but merely to write the word short, thus, is simil." Moreover, in the Preface he is informed that "more competent judges than the editor" advised that the symbol be eliminated, and so it was, except in one or two instances where "it was too late to make the alteration," the sheets having already been printed. Of course, one reason for failure to adopt ^ for "similar" in England lies in the fact that ^ was used there for "difference." 373. When the sides of the triangle ABC and A'B'C' are con- sidered as being vectors, special symbols have been used by some authors to designate different kinds of similarity. Thus, Stolz and Grneiner 2 employ ^ to mark that the similar triangles are uniformly similar (einstimmig dhnlich), that is, the equal angles of the two tri- angles are all measured clockwise, or all counter-clockwise; they em- ploy ^ to mark that the two triangles are symmetrically similar, that is, of two numerically equal angles, one is measured clockwise and the other counter-clockwise. The sign ^ has been used also for "is [or are] measured by," by Alan Sanders; 3 the sign 9= is used for "equals approximately," by Hudson and Lipka. 4 A. Pringsheim 5 uses the symbolism a v ^ab v to express that , ^r^ a. 1 v = + QO b v 374. The sign ^ for congruence was not without rivals during the nineteenth century. Occasionally the sign =, first introduced by Riernann 6 to express identity, or non-Gaussian arithmetical congru- ence of the type (a+6) 2 = a 2 +2a6+6 2 , is employed for the expression of geometrical congruence. One finds == for congruent in W. Bolyai, 7 1 Elements of Euclid .... from the Text of Dr. Simson. By a Member of the University of Cambridge (London, 1827), p. 104. 2 O. Stolz und J. A. Gmeiner, Theoretische Arithmetik (Leipzig), Vol. II (1902), p. 332. 3 Alan Sanders, Plane and Solid Geometry (New York, [1901]), p. 14. 4 R. G. Hudson and J. Lipka, Manual of Mathematics (New York, 1917), p. 68. 5 A. Pringsheim, Mathematische Annalen, Vol. XXXV (1890), p. 302; En- cyclopedie des scien. Math., Tom. I, Vol. I (1904), p. 201, 202. 6 See L. Kronecker, Vorlesungen liber Zahlentheorie (Leipzig, 1901), p. 86; G. F. B. Riemann, Elliptische Funktiown (Leipzig, 1899), p. 1, 6. 7 W. Bolyai, Tentamen (2d ed.), Tom. I (Budapest, 1897), p. xi. 418 A HISTORY OF MATHEMATICAL NOTATIONS II . G. Grassrnann, 1 Dupuis, 2 Biulden, 3 Veronese, 4 Casey, 5 Halsted, 8 Baker, 7 Betz and Webb, 8 Young and Schwarz, 9 McDougall. 10 This sign = for congruence finds its widest adoption in Great Britain at the present time. Jordan 11 employs it in analysis to express equivalence. The idea of expressing similarity by the letter $ placed in a horizontal position is extended by Callet, who uses w, O, Q, to express "similar," "dissimilar," "similar or dissimilar/' 12 Callet's notation for "dissimilar" did not meet with general adoption even in his own country. The sign = has also other uses in geometry. It is used in the Riemaimian sense of "identical to," not "congruent," by Busch and Clarke, 13 Meyers, 14 E. R. Smith, 15 Wells and Hart. 16 The sign = or >< is made to express "equivalent to" in the Geometry of Hopkins. 17 The symbols ~ and ^ for "similar" have encountered some com- petition with certain other symbols. Thus "similar" is marked ||| in the geometries of Budden 18 and McDougall. The relation "coincides with," which Leibniz had marked with |~|, is expressed by = in White's Geometry. Cremona- denotes by I H. G. Grassmann in C relic's Journal, Vol. XLII (1851), p. 193-203. - N. F. Dupuis, Elementary Synthetic Geometry (London, 1899), p. 29. 3 E. Budden, Elementary Pure Geometry (London, 1904), p. 22. 4 Guiseppe Veronese, Elementi di Geomelria, Part I (3ded.; Verona, 1904), p. 11. 5 J. Casey, First Six Books of Euclid' 's Elements (7th ed.; Dublin, 1902). 6 G. B. Halsted, Rational Geometry (New York, 1904), p. vii. 7 Alfred Baker, Transactions of the Royal Society of Canada (2d ser., 1906-7), Vol. XII, Sec. Ill, p. 120. 8 W. Betz and H. E. Webb, Plane Geometry (Boston, [1912]), p. 71. a John W. Young and A. J. Schwartz, Plane Geometry (New York, [1905]). 10 A. II. McDougall, The Ontario High School Geometry (Toronto, 1914), p. 158. II Camille Jordan, Cours (V analyse, Vol. II (1894), p. (314. 12 Francois Callet, Tables portaiires de loyarilhmes (Paris, 1795), p. 79. Taken from Desire Andre, Notations ?nathcma(iques (Paris, 1909), p. 150. 13 W. N. Busch and John B. Clarke, Elements of Geometry (New York, [1905]). 14 G. W. Meyers, Second-Year Mathematics for /Secondary Schools (Chicago, 1910), p. 119. 16 Eugene R. Smith, Solid Geometry [1913]. 18 W. Wells and W. W. Hart, Plane and Solid Geometry (Boston, [1905]), p. x. 17 Irving Hopkins, Manual of Plane Geometry (Boston, 1891), p. 10. 18 E. Budden, Elementary Pure Geometry (London, 1904), p. 22. 10 Emerson E. White, Elements of Geometry (New York City, .1895). 20 Luigi Cremona, Projectiue Geometry (trans. Oh. Leudesdorf; 2d ed.; Oxford, 1893), p. 1. GEOMETRY 419 a.BC^A' that the point common to the plane a and the straight line BC coincides with the point A'. Similarly, a German writer 1 of 1851 indicates by a=6, A = B that the two points a and b or the two straights A and B coincide (zusammenf alien) . 375. The sign =O for equivalence. In many geometries congruent figures are marked by the ordinary sign of equality, . To distin- guish between congruence of figures, expressed by =, and mere equivalence of figures or equality of areas, a new symbol O came to be used for "equivalent to" in the United States. The earliest appear- ance of that sign known to us is in a geometry brought out by Charles Davies 2 in 1851. He says that the sign "denotes equivalency and is read is equivalent to." The curved parts in the symbol, as iised by Davies, are not semicircles, but semiellipses. The sign is given by Davies and Peck, 8 Benson, 4 Wells, 5 Went worth, 6 McDonald, 7 Macnie, 8 Phillips and Fisher, 9 Milne, 10 McMahon, 11 Durcll, 12 Hart and Feldman. 13 It occurs also in the trigonometry of Anderegg and Roe. 14 The signs =0= and = for equivalence and equality